Mds array codes with optimal building

ABSTRACT

MDS array codes are widely used in storage systems to protect data against erasures. The rebuilding ratio problem is addressed and efficient parity codes are proposed. A controller as disclosed is configured for receiving configuration data at the controller that indicates operating features of the array and determining a parity code for operation of the array according to a permutation, wherein the configuration data specifies the array as comprising nodes defined by A=(a i,j ) with size r m ×k for some integers k, m, and wherein for T={v 0 , . . . , v k−1 } Z r   m  a subset of vectors of size k, where for each v=(v 1 , . . . , v m )εT, gcd(v 1 , . . . , v m , r), where gcd is the greatest common divisor, such that for any l, 0≦l≦r−1, and vεT, the code values are determined by the permutation f v   l :[0,r m −1]→[0,r m −1] by f v   l (x)=x+lv.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a non-provisional patent application and claims the benefit of co-pending U.S. Provisional Application No. 61/452,863 filed Mar. 15, 2011 entitled “MDS Array Codes with Optimal Building” and claims the benefit of co-pending U.S. Provisional Application No. 61/490,503 filed May 26, 2011 entitled “MDS Array Codes with Optimal Building”. The disclosures of both provisional applications are hereby expressly incorporated by reference in their entirety for all purposes.

STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under ECCS0802017 awarded by the National Science Foundation. The government has certain rights in the invention.

BACKGROUND

Data storage system with extremely large capacity have become commonplace and are typically structured as Redundant Arrays of Inexpensive Disks, or RAID. The fundamental structure of a RAID storage system is a collection of multiple disks (also referred to as drives) that are organized as an array in which read and write operations are performed under management of a RAID controller. If a disk drive in the array fails, causing an erasure, then the RAID controller can rebuild the erased data from the remaining disks. The RAID storage system stores data that it receives data from an external machine, referred to as a host computer, and retrieves data for delivery to a host computer, upon request. To the host computer, the RAID storage system appears to be a single logical hard disk.

The RAID controller manages the disks of the array according to configuration data received from a user, typically through a host computer. The configuration data includes, for example, the RAID level, how many disks are in the array, the drive identification names or numbers, the location of the data, and any other data required by the RAID controller to configure and manage the read, write, and rebuilding operations for the RAID array.

The disks in a RAID storage system include disks that contain data that a user wants to store and retrieve, called systematic data, and also include disks that contain redundancy information, called parity data, from which the RAID controller can rebuild any failed disks and recover from an erasure. The parity data is typically generated as a linear combination of data from the systematic disks, comprising redundancy values computed by calculating a function of predetermined word sizes from the data of the systematic disks. The parity data is stored on the parity disks. If a systematic disk fails, the data on the failed disk can be regenerated from the data stored on the parity drives and the remaining systematic disks. If a parity disk fails, its parity data can be reconstructed from the data stored on the systematic disks. Various codes have been proposed to provide efficient rebuilding of data upon suffering an erasure, but many conventional codes require relatively complicated or time-consuming parity computations. For example, Maximum Distance Separable (MDS) codes are proposed, as described further below, and provide improved storage efficiency.

Efficient codes are becoming more and more important as the size of RAID storage systems increases. For example, RAID storage systems may be structured so that the storage disks of the array are, in fact, nodes of a network, and not just single disk drives. That is, a RAID array may be comprised of network nodes, each of which may comprise a RAID array. Such high-density data storage systems urgently need more efficient storage codes. It is desired to improve the efficiency of RAID codes for read and write operations and rebuilding operations in RAID arrays.

Other features and advantages of the present disclosure should be apparent from the following description of exemplary embodiments, which illustrate, by way of example, aspects of the disclosure.

SUMMARY

The rebuilding ratio for multiple-node storage systems is improved. A controller as disclosed is configured for receiving configuration data at the controller that indicates operating features of the array and determining a parity code for operation of the array according to a permutation, wherein the configuration data specifies the array as comprising nodes defined by A=(a_(i,j)) with size r^(m)×k for some integers k, m, and wherein for T={v₀, . . . , v_(k−1)}

Z_(r) ^(m) a subset of vectors of size k, where for each v=(v₁, . . . , v_(m))εT, gcd(v₁, . . . , v_(m), r), where gcd is the greatest common divisor, such that for any l, 0≦l≦r−1, and vεT, the code values are determined by the permutation f_(v) ^(l):[0, r^(m)−1]→[0, r^(m)−1] by f_(v) ¹(x)=x+lv.

Other features and advantages of the present invention should be apparent from the following description of exemplary embodiments, which illustrate, by way of example, aspects of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagrammatic representation of rebuilding a (4, 2) MDS array code.

FIG. 2 is a diagrammatic representation of permutations for zigzag sets in a (5, 3) code configured with four rows.

FIG. 3( a) shows a set of orthogonal permutations as in Theorem 8 with sets X₀={0, 3}, X₁={0, 1}, X₂={0, 2}.

FIG. 3( b) shows a (5, 3) MDS array code generated by the orthogonal permutations.

FIG. 4 shows a 2-duplication of the code in FIG. 3( a) and FIG. 3( b).

FIG. 5 shows a (5, 3) Zigzag code generating from a standard basis vector and the zero vector.

FIG. 6 shows an erroneous array of the (5, 3) Zigzag code of FIG. 5.

FIG. 7 shows a (6, 3) MDS array code with optimal ratio 1/3.

FIG. 8 shows a block diagram of a RAID storage system constructed in accordance with the disclosure.

FIG. 9 shows a flow diagram of the operations performed by the RAID controller of FIG. 8.

FIG. 10 shows a block diagram of a computer system such as utilized as the RAID controller or host computer of FIG. 8.

DETAILED DESCRIPTION

The contents of this Detailed Description are organized according to the following headings:

I. Introduction

II. Definitions and Problem Settings

III. (k+2, k) MDS Array Code Constructions

IV. Code Duplication

V. Finite Field Size of a Code

VI. Decoding of the Codes

VII. Generalization of the Code Construction

VIII. Rebuilding Multiple Erasures

IX. Physical Implementation

X. Concluding Remarks

Equations will be identified by a parenthetical numeral at the right end of a line of text, such as (1). Subheadings are not listed in this outline of headings.

I. INTRODUCTION

Erasure-correcting codes are the basis of the ubiquitous RAID schemes for storage systems, where disks correspond to symbols in the code. Specifically, RAID schemes are based on MDS (maximum distance separable) array codes that enable optimal storage and efficient encoding and decoding algorithms. With r redundancy symbols, an MDS code is able to reconstruct the original information if no more than r symbols are erased. An array code is a two dimensional array, where each column corresponds to a symbol in the code and is stored in a disk in the RAID scheme. We are going to refer to a disk/symbol as a node or a column interchangeably, and an entry in the array as an element. Examples of MDS array codes may be found in documents describing MDS array codes such as “EVENODD” see M. Blaum et al., IEEE Trans. on Computers, 44(2):192-202 (February 1995); and M. Blaum et al., IEEE Trans. on Information Theory, 42:529-542 (1996); “B-code” see L. Xu et al., IEEE Trans. on Information Theory, 45(6):1817-1826 (September 1999)3, “X-code” see L. Xu and J. Bruck, IEEE Trans. on Information Theory, 45(1):272-276 (1999), “RDP” see P. Corbett et al., in Proc. of the 3rd USENIX Symposium on File and Storage Technologies (FAST 04) (2004)5, and “STAR-code” see C. Huang and L. Xu, IEEE Trans. on Computers, 57(7):889-901 (2008).

Suppose that some nodes are erased in an systematic MDS array code. We will rebuild them by accessing (reading) some information in the surviving nodes, all of which are assumed to be accessible. The fraction of the accessed information in the surviving nodes is called the rebuilding ratio. If r nodes are erased, then the rebuilding ratio is 1, because we need to read all the remaining information. However, is it possible to lower this ratio for less than r erasures? For example, FIG. 1 shows the rebuilding of the first systematic (information) node for an MDS code with four information elements and two redundancy nodes, which requires the transmission (reading) of three elements. If the first column containing elements a and c is erased, then reading the three circled elements can provide the original elements a and c. That is, following the failure of the two elements a, c in the first column, it will be necessary to read three of the remaining six elements in the array. Thus, the rebuilding ratio is 1/2. In a storage system there is a difference between erasures of systematic and parity nodes. Erasure of the former will affect the information access time since part of the raw information is missing, however erasure of the latter does not have such effects, since the entire raw information is accessible. Moreover in most storage systems the number of parity nodes is negligible compared to the systematic ones. Therefore our construction focuses on optimally rebuilding the systematic nodes.

In documents such as A. Dimakis et al., IEEE Trans. on Information Theory, 56(9):4539-4551 (2010), and Y. Wu et al., in Allerton Conference on Control, Computing, and Communication, Urbana-Champaign, Ill. (2007), a related problem is discussed: the nodes are assumed to be distributed and fully connected in a network, and the repair bandwidth is defined as the minimum amount of data needed to transmit in the network in order to retain the MDS property. Note that one block of data transmitted can be a function of several blocks of data. In addition, retaining MDS property does not imply rebuilding the original erased node, whereas we restrict our problem to exact rebuilding. Therefore, the repair bandwidth is a lower bound of the rebuilding ratio.

An (n, k) MDS code has n nodes in each codeword and contains k nodes of information and r=n−k nodes of redundancy. A lower bound for the repair bandwidth was shown as [A. Dimakis et al., IEEE Trans. on Information Theory, 56(9):4539-4551 (2010)]

$\begin{matrix} {{\frac{M}{k} \cdot \frac{n - 1}{n - k}},} & (1) \end{matrix}$

where M is the total amount of information, and all the surviving nodes are assumed to be accessible. It can be verified that FIG. 1 matches this lower bound. A number of works addressed the repair bandwidth problem (see A. Dimakis et al., IEEE Trans. on Information Theory, 56(9):4539-4551 (2010); Y. Wu et al., in Allerton Conference on Control, Computing, and Communication, Urbana-Champaign, Ill. (2007); Y. Wu, in ISIT (2009); Y. Wu and A. Dimakis, in ISIT (2009); K. V. Rashmi et al., Tech Rep. arXiv:0906.4913 (2009); C. Suh and K. Ramchandran, Tech. Rep. arXiv:1001.0107 (January 2010); N. B. Shah et al., in IEEE Information Theory Workshop (ITW) (January 2010); C. Suh and K. Ramchandran, Tech. Rep. arXiv:1004.4663 (2010); V. R. Cadambe et al., Tech. Rep. arXiv:1004.4299 (2010); K. V. Rashmi et al., Tech. Rep. arXiv:1101.0133 (2010), and it was shown by interference alignment in documents such as C. Suh and K. Ramchandran, Tech. Rep. arXiv:1001.0107 (January 2010), N. B. Shah et al., in IEEE Information Theory Workshop (ITW) (January 2010) that this bound is asymptotically achievable for exact repair. Instead of trying to construct MDS codes that can be easily rebuilt, a different approach [see Z. Wang et al., in IEEE GLOBECOM Workshops, 1905-1909 (December 2010), and L. Xiang et al., in ACM SIGMETRICS, 119-130 (2010)] was used by trying to find ways to rebuild existing families of MDS array codes. The ratio of rebuilding a single systematic node was shown to be 3/4+o(1) for EVENODD or RDP codes, both of which have two parities. However, from the lower bound of (1) the ratio is 1/2. A related work for optimal rebuilding problem can be found in V. R. Cadambe et al., in ISIT (2011).

The main contribution of the present discussion is that it provides the first explicit construction of systematic (n,k) MDS array codes for any constant r=n−k, which achieves optimal rebuilding ratio of 1/r. Moreover, they also achieve optimal ratio of e/r when e systematic erasures occur, 1≦e≦r. We call them intersecting zigzag sets codes (IZS codes). The parity symbols are constructed by linear combinations of a set of information symbols, such that each information symbol is contained exactly once in each parity node. These codes have a variety of advantages: (1) they are systematic codes, and it is easy to retrieve information; (2) they have high code rate k/n, which is commonly required in storage systems; (3) the encoding and decoding of the codes can be easily implemented (for r=2, the code uses finite field of size three); (4) they match the lower bound of the ratio when rebuilding e systematic nodes; (5) the rebuilding of a failed node requires simple computation and access to only 1/r of the data in each node (no linear combination of data); and 6) they have optimal update, namely, when an information element is updated, only n−k+1 elements in the array need update.

The remainder of the paper is organized as follows. Section II gives definitions and some general observations on MDS array codes, and Section III constructs (k+2, k) MDS array codes with optimal rebuilding ratio. Section IV introduces code duplication and thus generates (k+2, k) MDS array codes for arbitrary number of columns. We discuss the size of the finite field needed for these constructions in Section V. Decoding in case of erasure and errors are discussed in VI. Section VII generalizes the MDS code construction to arbitrary number of parity columns. These generalized codes have similar properties as the (k+2, k) MDS array codes, likewise some of them has optimal ratio. Rebuilding of multiple erasures and generalization of the rebuilding algorithms are shown in Section VIII. Examples of physical implementations of the principles disclosed herein are shown in Section IX. Finally we provide concluding remarks in Section X.

II. DEFINITIONS AND PROBLEM SETTINGS

In the rest of this document, we are going to use [i, j] to denote {i, i+1, . . . , j} and [i] to denote {1, 2, . . . , j}, for integers i≦j. And denote the complement of a subset X

M as X=M\X. For a matrix A, A^(T) denotes the transpose of A. For a binary vector v=(v₁, . . . , v_(n)) we denote by v=(v₁+1 mod 2, . . . , v_(n)+1 mod 2) its complement vector. The standard vector basis of dimension m will be denoted as {e_(i)}_(i=1) ^(m) and the zero vector will be denoted as e₀. For an integer n denote by S_(n) the set of permutations over the integers [0, n−1]. For two functions f and g, denote their composition by fg or f∘g.

Let us define a MDS array code with two parities. Let A=(a_(i,j)) be an array of size p×k over a finite field F, where iε[0, p−1], jε[0,k−1], and each of its entry is an information element. We add to the array two parity columns and obtain an (n=k+2, k) MDS code of array size p×n. Each element in these parity columns is a linear combination of elements from A. More specifically, let the two parity columns be C_(k)(r₀, r₁, . . . , r_(p−1))^(T) and C_(k+1) (z₀, z₁, . . . , z_(p−1))^(T). Let R=, {R₀, R₁, . . . , R_(p−1)} and Z={Z₀, Z₁, . . . , Z_(p−1)} be two sets such that R₁, Z₁ are subsets of elements in A for all lε[0, p−1]. Then for all lε[0, p−1], define r_(l)=Σ_(aεR) ₁ α_(a)a and z₁=Σ_(aεZ) ₁ β_(a)a, for some sets of coefficients {α_(a)}, {β_(a)}

F. We call R and Z as the sets that generate the parity columns.

We assume the code has optimal update, meaning that only 3 elements in the code are updated when an information element is updated. Under this assumption, the following theorem characterizes the sets R and Z.

Theorem 1.

For an (k+2, k) MDS code with optimal update, the sets R and Z are partitions of A into p equally sized sets of size k, where each set in R or Z contains exactly one element from each column.

Proof: Since the code is a (k+2, k) MDS code, each information element should appear at least once in each parity column C_(k), C_(k+1). However, since the code has optimal update, each element appears exactly once in each parity column.

Let XεR, note that if X contains two entries of A from the systematic column C_(i), iε[0,k−1], then rebuilding is impossible if columns C_(i) and C_(k+1) are erased. Thus X contains at most one entry from each column, therefore |X|≦k. However each element of A appears exactly once in each parity column, thus if |X|<k, XεR, there is YεR, with |Y|>k, which leads to a contradiction since Y contains two elements from the same column. Therefore, |X|=k for all XεR. As each information element appears exactly once in the first parity column, R={R₀, . . . , R_(p−1)} is a partition of A into p equally sized sets of size k. Similar proof holds for the sets Z={Z₀, . . . , Z_(p−1)}.

By the above theorem, for the j-th systematic column (a₀, . . . , a_(p−1))^(T), its p elements are contained in p distinct sets R_(l), lε[0, p−1]. In other words, the membership of the j-th column's elements in the sets {R_(l)} defines a permutation g_(j):[0, p−1]→[0, p−1], such that g_(j)(i)=l iff a_(i)εR_(l). Similarly, we can define a permutation f_(j) corresponding to the second parity column, where f_(j)(i)=l iff a_(i)εZ_(l). For example, FIG. 2 is a diagrammatic representation 200 (in a table format) of permutations for zigzag sets in a (5, 3) code with four rows. Columns 0, 1, and 2 are systematic nodes containing data and columns R and Z are parity nodes. Each element in column R is a linear combination of the systematic elements in the same row, i.e., to the left. Each element in column Z is a linear combination of the systematic elements with the same symbol. The shaded elements are accessed to rebuild column 1. That is, each element in the parity column Z is a linear combination of elements with the same symbol. For instance the

in column Z is a linear combination of all the

elements in columns 0, 1, and 2. And each systematic column corresponds to a permutation of the four symbols.

Observing that there is no importance of the elements' ordering in each column, w.l.o.g. we can assume that the first parity column contains the sum of each row of A and g_(j)'s correspond to identity permutations, i.e. r_(i)=Σ_(j=0) ^(k−1)α_(i,j)α^(i,j) for some coefficients α_(i,j). We refer to the first and the second parity columns as the row column and the zigzag column respectively, likewise R_(l) and Z_(l), lε[0, p−1], are referred to as row sets and zigzag sets respectively. We will call f_(j), jε[0,k−1], zigzag permutations. By assuming that the first parity column contains the row sums, the code is uniquely defined by the following:

(1) The coefficients in the linear combinations.

(2) The zigzag permutations.

Then for our rebuilding problem, the desired properties of the code are:

(1) MDS.

(2) Minimized rebuilding ratio when one systematic node is erased.

Our approach consists of two steps: first we choose the appropriate zigzag sets Z₀, . . . Z_(p−1) in order to minimize the rebuilding ratio, and then we choose the coefficients in the linear combinations in order to make sure that the code is indeed an MDS code. But first we show that any set of zigzag sets Z={Z₀, . . . , Z_(p−1)} defines a (k+2, k) MDS array code over a field F large enough. For that proof we use the well-known Combinatorial Nullstellensatz by Alon [see N. Alon, Combinatorics Probability and Computing, 8(1-2):7-29 (January 1999)], given by Theorem 2:

Theorem 2.

(Combinatorial Nullstellensatz) Let F be an arbitrary field, and let f=f(x₁, . . . , x_(q)) be a polynomial in F[x₁, . . . , x_(q)]. Suppose the degree off is deg(f)=Σ_(i=1) ^(q)t_(i), where each t_(i) is a nonnegative integer, and suppose the coefficient of Π_(i=1) ^(q)x_(i) ^(t) ^(i) in f is nonzero. Then, if S₁, . . . , S_(n) are subsets of F with |S_(i)|>t_(i), there are s₁εS₁, s₂εS₂, . . . , s_(q)εS_(q) so that

f(s ₁ , . . . , s _(q))≠0.

Theorem 3.

Let A=(a_(i,j)) be an array of size p×k and the zigzag sets be Z={Z₀, . . . , Z_(p−1)}, then there exists a (k+2, k) MDS array code for A with Z as its zigzag sets over the field F of size greater than p(k−1)+1.

Proof: Assume the information of A is given in a column vector W of length pk, where column iε[0,k−1] of A is in the row set [(ip+1, (i+1)p] of W. Each systematic node i, iε[0,k−1], can be represented as Q_(i)W where Q_(i)=└0_(p×pi), I_(p×p), 0^(p×p(k−i−1))┘. Moreover define Q_(k)=└I_(p×p)I_(p×p), . . . , I_(p×p)┘, Q_(k+1)=[x₀P₀, x₁P₁, . . . , x_(k−1)P_(k−1)] where the P_(i)'s are permutation matrices (not necessarily distinct) of size p×p, and the x_(i)'s are variables, such that C_(k)=Q_(k)W, C_(k+1)=Q_(k+1)W. The permutation matrix P_(i)=(p_(l,m) ^((i))) is defined as p_(l,m) ^((i))=1 if and only if a_(m,i)εZ_(l). In order to show that there exists such MDS code, it is sufficient to show that there is an assignment for the intermediates {x_(i)} in the field F, such that for any set of integers {s₁, s₂, . . . , s_(k)}

[0, k+1] the matrix Q=[Q_(s) _(l) ^(T), Q_(s) _(l) ^(T), . . . , Q_(s) _(k) ^(T)] is of full rank. It is easy to see that if the parity column C_(k+1) is erased, i.e., k+1∉{s₁, s₂, . . . , s_(k)} then Q is of full rank. If k∉{s₁, s₂, . . . , s_(k)} and k+1ε{s₁, s₂, . . . , s_(q)} then Q is of full rank if none of the x_(i)'s equals to zero. The last case is when both k, k+1ε{s₁, s₂, . . . , s_(k)}, i.e., there are 0≦i≦j≦k−1 such that i, j∉{s₁, s₂, . . . , s_(k)}. It is easy to see that in that case _(Q) is of full rank if and only if the submatrix

$B_{i,j} = \begin{pmatrix} {x_{i}P_{i}} & {x_{j}P_{j}} \\ I_{p \times p} & I_{p \times p} \end{pmatrix}$

is of full rank. This is equivalent to det(B_(i,j))≠0. Note that deg(det(B_(i,j)))=p and the coefficient of x_(i) ^(p) is det(P_(i))ε{1,−1}. Define the polynomial

${T = {{T\left( {x_{0},x_{1},\ldots \;,x_{k - 1}} \right)} = {\prod\limits_{0 \leq i < j \leq {k - 1}}^{\;}\; {\det \left( B_{i,j} \right)}}}},$

and the result follows if there are elements a₀, a₁, . . . , a_(k−1)εF such that T(a₀, a₁, . . . , a_(k−1))≠0. T is of degree p(₂ ^(k)) and the coefficient of Π_(i=0) ^(k−1)x_(i) ^(p(k−1−i)) is Π_(i=0) ^(k−1)det(P_(i))^(k−1−i)≠0. Set for any i, S_(i)=F\0 in Theorem 2, and the result follows.

The above theorem states that there exist coefficients such that the code is MDS, and thus we will focus first on finding proper zigzag permutations {f_(i)}. The idea behind choosing the zigzag sets is as follows: assume a systematic column (a₀, a₁, . . . , a_(p−1))^(T) is erased. Each element a_(i) is contained in exactly one row set and one zigzag set. For rebuilding of element a_(i), access the parity of its row set or zigzag set. Moreover access the values of the remaining elements in that set, except a_(i). We say that an element a_(i) is rebuilt by a row (zigzag) if the parity of its row set (zigzag set) is accessed. For example, in FIG. 2, supposing column 1 is erased, one can access the shaded elements and rebuild its first two elements by rows, and the rest by zigzags. The set S={S₀, S₁, . . . , S_(p−1)} is called a rebuilding set for column (a₀, a₁, . . . , a_(p−1))^(T) if for each i, S_(i)εR∪Z and a_(i)εS_(i). In order to minimize the number of accesses to rebuild the erased column, we need to minimize the size of

|∪_(i=0) ^(p−1) S _(i)|,  (2),

which is equivalent to maximizing the number of intersections between the sets {S_(i)}_(i=0) ^(p−1). More specifically, the intersections between the row sets in S and the zigzag sets in S. For the rebuilding of node i by S, define the number of intersections by

$I\left( {{i\left. S \right)} = {{{\sum\limits_{S \in S}^{\;}{S}} - {{\bigcup_{S \in S}S}}} = {{pk} - {{{U_{S \in S}S}}.}}}} \right.$

Moreover define the number of total intersections in an MDS array code C as

${I(C)} = {\sum\limits_{i = 0}^{k - 1}{s{\max\limits_{\mspace{14mu} {{rebuilds}\mspace{14mu} i}}\; {I\left( {i{\left. S \right).}} \right.}}}}$

Now define h(k) to be the maximal possible intersections over all (k+2, k) MDS array codes, i.e.,

${h(k)} = {\max\limits_{C}\mspace{11mu} {{I(C)}.}}$

For example, in FIG. 2 the rebuilding set for column 1 is S={R₀, R₁, Z₀, Z₁}, the size in equation (2) is 8, and I(1|S)=4. Note that each surviving node accesses exactly one-half of its information without performing any calculation within it.

The following theorem gives a recursive bound for the maximal number of intersections.

Theorem 4.

Let q≦k≦p, then

${h(k)}{\frac{{k\left( {k - 1} \right)}{h(q)}}{q\left( {q - 1} \right)}.}$

Proof: Let A be an information array of size p×k. Construct a MDS array code C by the row sets and the zigzag sets that reaches the maximum possible number of intersections, and suppose S^(i) achieves the maximal number of intersections for rebuilding column i, iε[0,k−1]. Namely the zigzag sets Z of the code C and the rebuilding sets S^(i) satisfy that,

${h(k)} = {{I(C)} = {\sum\limits_{i = 0}^{k - 1}{\max\limits_{s\mspace{11mu} {rebuilds}\mspace{11mu} i}{I\left( {{i\left. S \right)} = {\sum\limits_{i = 0}^{k - 1}{I\left( {i{\left. S^{i} \right).}} \right.}}} \right.}}}}$

For a subset of columns T

[0,k−1] and a rebuilding set S^(i)={S₀, . . . , S_(p−1)} we define the restriction of S^(i) to T by S_(T) ^(i)={S_(0,T), . . . , S_(p−1,T)}, which are the sets of elements in columns T. Denote by

${I\left( {j,S^{i}} \right)} = {{\sum\limits_{i = 0}^{p - 1}{{S_{i}\bigcap j}}} - {{\left( {\bigcup_{i = 0}^{p - 1}S_{i}} \right)\bigcap j}}}$

the number of intersections in column j while rebuilding column i by S^(i). It is easy to see that

${I\left( i \middle| S^{i} \right)} = {\sum\limits_{{j\text{:}j} \neq 1}\; {I\left( {j,S^{i}} \right)}}$

and thus

${h(k)} = {\sum\limits_{i,{{j\text{:}j} \neq i}}\; {{I\left( {i,S^{i}} \right)}.}}$

Note also that if i≠j and i, jεT, then Equation (3) gives:

I(j,S ^(i))=I(j,S _(T) ^(i)).  (3)

Hence

$\begin{matrix} {{\begin{pmatrix} {k - 2} \\ {q - 2} \end{pmatrix}{h(k)}} = {\begin{pmatrix} {k - 2} \\ {q - 2} \end{pmatrix}{\sum\limits_{{i,{j\text{:}}}{j \neq i}}\; {I\left( {j,S^{i}} \right)}}}} \\ {= {\sum\limits_{{i,{j\text{:}}}{j \neq 1}}^{\;}\; {\sum\limits_{{T \subseteq {{\lbrack{0,{k - 1}}\rbrack}\text{:}}}{i,{j \in T},{{|T|} = q}}}\; {I\left( {j,S^{i}} \right)}}}} \\ {= {\sum\limits_{{i,{j\text{:}}}{j \neq i}}{\sum\limits_{{T \subseteq {{\lbrack{0,{k - 1}}\rbrack}\text{:}}}{i,{j \in T},{{|T|} = q}}}\; {I\left( {j,S_{T}^{i}} \right)}}}} \\ {= {\sum\limits_{{{T \subseteq {{\lbrack{0,{k - 1}}\rbrack}\text{:}}}|T|} = q}\; {\sum\limits_{{i,{j \in {T\text{:}}}}{i \neq j}}\; {I\left( {j,S_{T}^{i}} \right)}}}} \end{matrix}\begin{matrix} \begin{matrix} {\mspace{135mu} {\leqq {\sum\limits_{{{T \subseteq {\lbrack{0,{k - 1}}\rbrack}}|T|} = q}\; {h(q)}}}} \\ {\mspace{130mu} {= {\begin{pmatrix} k \\ q \end{pmatrix}{{h(q)}.}}}} \end{matrix} & (4) \end{matrix}$

Inequality (4) above holds because the code restricted in columns T is a (q+2, q) MDS and optimal-update code, and h(q) is the maximal intersections among such codes. Hence,

${{{h(k)} \leqq \frac{\begin{pmatrix} k \\ q \end{pmatrix}{h(q)}}{\begin{pmatrix} {k - 2} \\ {q - 2} \end{pmatrix}}} = \frac{{k\left( {k - 1} \right)}{h(q)}}{q\left( {q - 1} \right)}},$

and the result follows.

For a (k+2, k) MDS code C with p rows define R(C) as the rebuilding ratio as the average fraction of accesses in the surviving systematic and parity nodes while rebuilding one systematic node, i.e.,

${R(C)} = {\frac{k\left( {{p\left( {k - 1} \right)} - {I(C)} + p} \right)}{{p\left( {k + 1} \right)}k} = {1 - {\frac{{I(C)} + {pk}}{{p\left( {k + 1} \right)}k}.}}}$

Notice that in the two parity nodes, we access p elements because each erased element must be rebuilt either by row or by zigzag. Thus we have the term p in the above definition. Define the ratio function for all (k+2, k) MDS codes with p rows as

${{R(k)} = {{{\,_{C}^{\min}\mspace{14mu} R}(C)} = {1 - \frac{{h(k)} + {pk}}{{p\left( {k + 1} \right)}k}}}},$

which is the minimal average portion of the array needed to be accessed in order to rebuild one erased column.

Theorem 5.

R(k) is no less than 1/2 and is a monotone nondecreasing function.

Proof: Consider a (k+2, k) code with p rows and assume a systematic node is erased.

In order to rebuild it, p row and zigzag sets are accessed. Let x and p−x be the number of elements that are accessed from the first and the second parity respectively. W.l.o.g we can assume that

${x\left( {k - 1} \right)} \geqq \frac{p\left( {k - 1} \right)}{2}$

otherwise p−x would satisfy it. Each element of these x sets is a sum of a set of size k. Thus in order to rebuild the node, we need to access at least

${x \geqq \frac{p}{2}},$

elements in the k−1 surviving systematic nodes, which is at least half of the size of these nodes. So the number of intersections is no more than

$\frac{{pk}\left( {k - 1} \right)}{2}.$

Thus

$\begin{matrix} {{h(k)} \leqq {\frac{{pk}\left( {k - 1} \right)}{2}.}} & (5) \end{matrix}$

and the ratio function satisfies

${R(k)} = {{{1 - \frac{{h(k)} + {pk}}{{pk}\left( {k + 1} \right)}} \geqq {1 - \frac{\frac{{pk}\left( {k - 1} \right)}{2} + {pk}}{{pk}\left( {k + 1} \right)}}} = {\frac{1}{2}.}}$

So the rebuilding ratio is no less than 1/2.

From Theorem 4 we get the following in Equation (6),

$\begin{matrix} {{{{h\left( {k - 1} \right)} \leqq \frac{\left( {k + 1} \right){{kh}(k)}}{k\left( {k - 1} \right)}} = {\frac{\left( {k + 1} \right){h(k)}}{\left( {k - 1} \right)}.{Hence}}},\begin{matrix} {{R\left( {k + 1} \right)} = {1 - \frac{h\left( {k + 1} \right)}{{p\left( {k + 1} \right)}\left( {k + 2} \right)} - \frac{1}{k + 2}}} \\ {{\geqq {1 - \frac{h(k)}{{p\left( {k - 1} \right)}\left( {k + 2} \right)} - \frac{1}{k + 2}}}} \\ {{= {1 - \frac{{h(k)} + {p\left( {k - 1} \right)}}{{p\left( {k - 1} \right)}\left( {k + 2} \right)}}}} \\ {{\geqq {1 - \frac{{h(k)} + {pk}}{{pk}\left( {k + 1} \right)}}}} \\ {{{= {R(k)}},}} \end{matrix}} & (6) \end{matrix}$

where Equation (7) follows from Equation (5). Thus the ratio function is nondecreasing.

The lower bound of 1/2 in the previous theorem can be also derived from (1). For example, it can be verified that for the code in FIG. 2, all the three systematic columns can be rebuilt by accessing half of the remaining elements. Thus the code achieves the lower bound 1/2, and therefore R(3)=1/2. Moreover, we will see in Corollary 11 that R(k) is almost 1/2 for all k and p=2^(m), where l is large enough.

III. (k+2, k) MDS ARRAY CODE CONSTRUCTIONS

The previous section gave us a lower bound for the ratio function. The question is can we achieve it? If so, how? We know that each (k+2, k) MDS array code with row and zigzag columns is defined by a set of permutations f₀, . . . , f_(k−1). The following construction constructs a family of such MDS array codes. From any set T

F₂ ^(m), |T|=k, we construct a (k+2, k) MDS array code of size 2^(m)×(k+2). The ratio of the constructed code will be proportional to the size of the union of the elements in the rebuilding set in Equation (2). Thus we will try to construct such permutations and subsets that minimize the union. We will show that some of these codes have the optimal ratio of 1/2.

In this section all the calculations are done over F₂. By notation herein we use xε[0,2^(m)−1] both to represent the integer and its binary representation. It will be clear from the context which meaning is in use.

Construction 1.

Let A=(a_(i,j)) be an array of size 2^(m)×k for some integers k, m, and k≦2^(m). Let T

F₂ ^(m) be a subset of vectors of size k which does not contain the zero vector. For vεT we define the permutation f_(v):[0,2^(m)−1]→[0,2^(m)−1] by f_(v)(x)=x+v, where x is represented in its binary representation. One can check that this is actually a permutation. For example when m=2, v=(1,0)

f _((1,0))(3)=(1,1)+(1,0)=(0,1)=1.

One can check that the permutation f_(v) in vector notation is [2,3,0,1]. In addition, we define X_(v) as the set of integers x in [0,2^(m)−1] such that the inner product between their binary representation and v satisfies x·v=0, e.g., X_((1,0))={0,1}. The construction of the two parity columns is as follows: The first parity column is simply the row sums. The zigzag sets Z₀, . . . , Z₂ _(m) ⁻¹ are defined by the permutations {f_(v j):v_(j)εT} as a_(i,j)εZ_(l) if f_(v) _(j) (i)=1. We will denote the permutation f_(v) _(j) as f_(j). Assume column j is erased, and define S_(r)={a_(i,j):iεX_(j)} and S_(z)={a_(i,j):iεX_(j)}. Rebuild the elements in S_(r), by rows and the elements in S_(z) by zigzags.

Recall that by Theorem 3 this code can be an MDS code over a field large enough. The following theorem gives the ratio for Construction 1.

Theorem 6.

The code described in Construction 1 and generated by the vectors v₀, v₁, . . . , v_(k−1) is a (k+2, k) MDS array code with ratio

$\begin{matrix} {R = {\frac{1}{2} + {\frac{\left. {\sum\limits_{i = 0}^{k - 1}\; \sum\limits_{j \neq i}}\; \middle| {{f_{i}\left( X_{i} \right)}\bigcap{f_{j}\left( X_{i} \right)}} \right|}{2^{m}{k\left( {k + 1} \right)}}.}}} & (8) \end{matrix}$

Proof: In rebuilding of node i we rebuild the elements in rows X_(i) by rows, thus the row parity column accesses the values of the sum of rows X_(i). Moreover, each surviving systematic node accesses its elements in rows X_(i). Hence, by now |X_(i)|k=2^(m−1) k elements are accessed, and we manage to rebuild the elements of node i in rows X_(i). The elements of node i in rows x_(i) are rebuilt by zigzags, thus the zigzag parity column accesses the values of the zigzags sums {z_(f) _(i) _((l)):lε X_(i) }, and each surviving systematic node accesses the elements of these zigzags from its column, unless these elements are already included in the rebuilding by rows. The zigzag elements in {Z_(f) _(i) _((l)):lε X_(i) } of node j are in rows f_(j) ⁻¹(f_(i)( X_(i) )), thus the extra elements node j needs to access are in rows f_(j) ⁻¹(f_(i)( X_(i) ))\X_(i). But,

$\begin{matrix} {\left| {{f_{j}^{- 1}\left( {f_{i}\left( \overset{\_}{X_{i}} \right)} \right)}\backslash X_{i}} \right|} \\ {= \left| {\overset{\_}{f_{j}^{- 1}\left( {f_{i}\left( X_{i} \right)} \right)}\bigcap\overset{\_}{X_{i}}} \right|} \\ {= \left| {{f_{j}^{- 1}\left( {f_{i}\left( X_{i} \right)} \right)}\bigcup X_{i}} \right|} \\ {= \left. {2^{m} -} \middle| {{f_{j}^{- 1}\left( {f_{i}\left( X_{i} \right)} \right)}\bigcup X_{i}} \right|} \\ {= {2^{m} - \left( \left| {f_{j}^{- 1}\left( {f_{i}\left( X_{i} \right)} \right)} \middle| {+ \left| X_{i} \middle| {- \left| {{f_{j}^{- 1}\left( {f_{i}\left( X_{i} \right)} \right)}\bigcap X_{i}} \right|} \right.} \right. \right)}} \\ {= \left| {{f_{j}^{- 1}\left( {f_{i}\left( X_{i} \right)} \right)}\bigcap X_{i}} \right|} \\ {{= \left| {{f_{i}\left( X_{i} \right)}\bigcap{f_{j}\left( X_{i} \right)}} \right|},} \end{matrix}$

where we used the fact that f_(i), f_(j) are bijections, and |X_(i)|=2^(m−1). Hence in a rebuilding of node i the number of elements to be accessed is 2^(m−1)(k+1)+Σ_(j≠i)|(f_(i)(X_(i)))∩f_(j)(X_(i))|. The result follows by dividing by the size of the remaining array 2^(m)(k+1) and averaging over all systematic nodes.

The following lemma will help us to calculate the sum in Equation (8). Define |v\u|=Σ_(i:v) _(i) _(=1,u) _(i) ₌₀1 as the number of coordinates at which v has a 1 but u has a 0. Lemma 7 for any 0≠v, uεT

$\begin{matrix} {\left| {{f_{v}\left( X_{v} \right)}\bigcap{f_{u}\left( X_{v} \right)}} \right| = \left\{ \begin{matrix} {\left| X_{v} \right|,} & {\left| {v\backslash u} \middle| \mspace{14mu} {{mod}\mspace{14mu} 2} \right. = 0} \\ {0,} & {\left| {v\backslash u} \middle| \mspace{14mu} {{mod}\mspace{14mu} 2} \right. = 1.} \end{matrix} \right.} & (9) \end{matrix}$

Proof: Consider the group (F₂ ^(m), +). Recall that f_(v)(X)=X+v={x+v:xεX}. The sets f_(v)(X_(v))=X_(v)+v and f_(u)(X_(v))=X_(v)+u are cosets of the subgroup X_(v)={wεF₂ ^(m):w·v=0}, and they are either identical or disjoint. Moreover, they are identical iff v−uεX_(v), namely (v−u)·v=Σ_(i:v) _(i) _(=1,u) _(i) ₌₀1≡0 mod 2. However, by definition |v\u|≡Σ_(i:v) _(i) _(=1,u) _(i) ₌₀1 mod 2, and the result follows.

This construction enables us to construct an MDS array code from any subset of vectors in F₂ ^(m). However, it is not clear which subset of vectors should be chosen. The following is an example of a code construction for a specific set of vectors.

Example 1

Let T={vεF₂ ^(m):∥v∥₁=3} be the set of vectors with weight 3 and length m. Notice that |T|=(₃ ^(m)). Construct the code C by T according to Construction 1. Given vεT,

${{\left\{ {{u \in {T:{{v\backslash u}}}} = 3} \right\} } = \begin{pmatrix} {m - 3} \\ 3 \end{pmatrix}},$

which is the number of vectors with 1's in different positions as v. Similarly,

${\left\{ {{u \in {T:{{v\backslash u}}}} = 2} \right\} } = {3\begin{pmatrix} {m - 3} \\ 2 \end{pmatrix}}$

and |{uεT:|v\u|=1}|=3(m−3). By Theorem 6 and Lemma 7, for large m the ratio is

${\frac{1}{2} + \frac{2^{m - 1}\begin{pmatrix} m \\ 3 \end{pmatrix}3\begin{pmatrix} {m - 3} \\ 2 \end{pmatrix}}{2^{m}\begin{pmatrix} m \\ 3 \end{pmatrix}\left( {\begin{pmatrix} m \\ 3 \end{pmatrix} + 1} \right)}} \approx {\frac{1}{2} + {\frac{9}{2m}.}}$

Note that this code reaches the lower bound of the ratio as m tends to infinity. In the following we will construct codes that reach the lower bound exactly.

Let {f₀, . . . , f_(k−1)} be a set of permutations over the set [0,2^(m)−1] with associated subsets X₀, . . . , X_(k−1)

[0,2^(m)−1], where each |X_(i)=2^(m−1). We say that this set is a set of orthogonal permutations if for any i, jε[0,k−1] and

${\frac{\left| {{f_{i}\left( X_{i} \right)}\bigcap{f_{j}\left( X_{i} \right)}} \right|}{2^{m - 1}} = \delta_{i,j}},$

where δ_(i,j) is the Kronecker delta. The following theorem constructs a set of orthogonal permutations of size m+1 using the standard basis vectors and the zero vector.

Theorem 8.

The permutations f₀, . . . , f_(m) and sets X₀, . . . , X_(m) constructed by the vectors {e_(i)}_(i=0) ^(m) and Construction 1 where X₀ is modified to be X₀={xεF₂ ^(m):x·(1, 1, . . . , 1)=0} is a set of orthogonal permutations. Moreover the (m+3, m+1) MDS array code of array size 2^(m)×(m+3) defined by these permutations has optimal ratio of 1/2. Hence, R(m+1)=1/2.

Proof: Since |e_(i)\e_(j)=1 for any i≠j≠0, we get by Lemma 7 that

f _(i)(X _(i))∩f _(i)(X _(i))=Ø.

Note that f_(i)(X_(i))={x+e_(i):x·e_(i)=0}={y:y·e_(i)=1}, so

f _(i)(X _(i))∩f ₀(X _(i))={y:y·e _(i)=1}∩{x:x·e _(i)=0}=Ø.

Similarly, f_(i)(X₀)={x+e_(i):x·(1, 1, . . . , 1)=0}={y:y·(1, 1, . . . , 1)=1}, and

$\begin{matrix} {{{f_{0}\left( X_{0} \right)}\bigcap{f_{i}\left( X_{0} \right)}} = {\left\{ {{x\text{:}{x \cdot \left( {1,\ldots,1} \right)}} = 0} \right\}\bigcap\left\{ {{y\text{:}{y\left( {1,\ldots,1} \right)}} = 1} \right\}}} \\ {= {\varphi.}} \end{matrix}$

Hence the permutations f₀, . . . , f_(m) are orthogonal permutations and the ratio is 1/2 by Theorem 6.

Note that the optimal code can be shortened by removing some systematic columns and still retain an optimal ratio, i.e., for any k≦m+1 we get R(k)=1/2. Actually this set of orthogonal permutations is optimal in size, as the following theorem suggests.

Theorem 9.

Let F be an orthogonal set of permutations over the integers [0,2^(m)−1], then the size of F is at most m+1.

Proof: We will prove it by induction on m. For m=0 there is nothing to prove. Let F={f₀, f₁, . . . , f_(k−1)} be a set of orthogonal permutations over the set [0,2^(m)−1]. We only need to show that |f|=k≦m+1. It is trivial to see that for any g, hεS₂ _(m) set hFg={hf₀g, hf₁g, . . . , hf_(k−1)g} is also a set of orthogonal permutations with sets g⁻¹(X₀), g⁻¹(X₁), . . . , g⁻¹(X_(k−1)). Thus w.l.o.g. we can assume that f₀ is the identity permutation and X₀=[0,2^(m−1)−1]. From the orthogonality we get that

∪_(i=1) ^(k−1) f _(i)(X ₀)= X ₀ =[2^(m−1), 2^(m)−1].

We claim that for any i≠0,

$\left| {X_{i}\bigcap X_{0}} \right| = {\frac{\left| X_{0} \right|}{2} = {2^{m - 2}.}}$

Assume the contrary, thus w.l.o.g we can assume that |X_(i)∩X₀|>2^(m−2), otherwise |X_(i)∩ X₀ |>2^(m−2). For any j≠i≠0 we get that

f _(j)(X _(i) ∩X ₀),f _(i)(X _(i) ∩X ₀)

X ₀ ,  (10)

$\begin{matrix} {\left| {f_{j}\left( {X_{i}\bigcap X_{0}} \right)} \right| = {\left| {f_{i}\left( {X_{i}\bigcap X_{0}} \right)} \middle| {> 2^{m - 2}} \right. = {\frac{\left| \overset{\_}{X_{0}} \right|}{2}.}}} & (11) \end{matrix}$

From equations (10) and (11) we conclude that f_(j)(X_(i)∩X₀)∩f_(i)(X_(i)∩X₀)≠Ø, which contradicts the orthogonality property. Define the set of permutations F*={f_(i)*}_(i=1) ^(k−1) over the set of integers [0,2^(m−1)−1] by f_(i)*(x)=f_(i)*(x)−2^(m−1), which is a set of orthogonal permutations with sets {X_(i)∩X₀}_(i=1) ^(k−1). By induction k−1≦m and the result follows.

The above theorem implies that the number of rows has to be exponential in the number of columns in any systematic code with optimal ratio and optimal update. Notice that the code in Theorem 8 achieves the maximum possible number of columns, m+1. An exponential number of rows is practical in storage systems, since they are composed of dozens of nodes (disks) each of which has size in an order of gigabytes. In addition, increasing the number of columns can be done using duplication (Theorem 10) or a larger set of vectors (Example 1) with a cost of a small increase in the ratio.

Example 2

Let A be an array of size 4×3. We construct a (5, 3) MDS array code for A as in Theorem 8 that accesses 1/2 of the remaining information in the array to rebuild any systematic node (see FIG. 3). For example, X₁={0, 1}, and for rebuilding of node 1 (column C₁) we access the elements a_(0,0), a_(0,2), a_(1,0), a_(1,2), and the following four parity elements

r ₀ =a _(0,0) +a _(0,1) +a _(0,2)

r ₁ =a _(1,0) +a _(1,1) +a _(1,2)

z _(f) ₁ ₍₂₎ =z ₀ =a _(0,0)+2a _(2,1)+2a _(1,2)

z _(f) ₁ ₍₃₎ =z ₁ +a _(1,0)+2a _(3,1) +a _(0,2).

It is trivial to rebuild node 1 from the accessed information. Note that each of the surviving node accesses exactly 1/2 of its elements. It can be easily verified that the other systematic nodes can be rebuilt the same way. With reference to FIG. 3( a) and FIG. 3( b), rebuilding a parity node is easily achieved by accessing all the information elements. FIG. 3( a) shows a set of orthogonal permutations as in Theorem 8, with sets X₀={0, 3}, X₁={0, 1}, X₂={0, 2}. FIG. 3( b) shows a (5, 3) MDS array code generated by the orthogonal permutations, as indicated by the arrow from FIG. 3( a) to FIG. 3( b). The first parity column C₃ is the row sum and the second parity column C₄ is generated by the zigzags. For example, zigzag z₀ contains the elements a_(i,j) that satisfy f_(j)(i)=0.

IV. CODE DUPLICATION

In this section, we are going to increase the number of columns in the constructed (k+2, k) MDS codes, such that k does not depend on the number of rows, and ratio is approximately 1/2.

Let C be a (k+2, k) array code with p rows, where the zigzag sets {Z_(l)}_(l=0) ^(p−1) are defined by the set of permutations {f_(i)}_(i=0) ^(k−1)

S_(p). For an integer s, an s-duplication code C′ is an (sk+2, sk) MDS code with zigzag permutations defined by duplicating the k permutations s times each, and the first parity column is the row sums. In order to make the code MDS, the coefficients in the parities may be different from the code C. For an s-duplication code, denote the column corresponding to the t-th f_(j) as column j^((t)), 0≦t≦s−1. Call the columns {j^((t)):jε[0,k−1]} the t-th copy of the original code. An example of a 2-duplication of the code in FIG. 3 is illustrated in FIG. 4. FIG. 4 shows that the code has six information (systematic) nodes, and two parity nodes, R and Z. The ratio, in accordance with Theorem 10 below, is 4/7.

Theorem 10.

If a (k+2, k) code C has ratio R(C), then its s-duplication code C′ has ratio

${R(C)} = {\left( {1 + \frac{s - 1}{{sk} + 1}} \right).}$

Proof: We propose a rebuilding algorithm for C′ with ratio of

${{R(C)} = \left( {1 + \frac{s - 1}{{sk} + 1}} \right)},$

which will be shown to be optimal.

Suppose in the optimal rebuilding algorithm of C, for column i, elements of rows J={j₁, j₂, . . . , j_(u)} are rebuilt by zigzags, and the rest by rows. In C′, all the s columns corresponding to f_(i) are rebuilt in the same way: the elements in rows J are rebuilt by zigzags.

-   -   W.l.o.g. assume column i⁽⁰⁾ is erased. Since column i^((t)),         tε[1, s−1] corresponds to the same zigzag permutation as the         erased column, for the erased element in the l-th row, no matter         if it is rebuilt by row or by zigzag, we have to access the         element in the l-th row and column i^((t)) (e.g. permutations f₀         ⁽⁰⁾, f₀ ⁽¹⁾ and the corresponding columns 0⁽⁰⁾, 0⁽¹⁾ in FIG. 4).         Hence all the elements in column i^((t)) must be accessed.         Moreover, the optimal way to access the other surviving columns         cannot be better than the optimal way to rebuild in the code C.         Thus the proposed algorithm has optimal rebuilding ratio.

When column i⁽⁰⁾ is erased, the average (over all iε[0,k−1]) of the number of elements needed to be accessed in columns l^((t)), for all lε[0,k−1], l≠1 and tε[0, s−1] is

R(C)p(k+1)−p.

Here the term −p corresponds to the access of the parity nodes in C. Moreover, we need to access all the elements in columns i^((t)), 0<t≦s−1, and access p elements in the two parity columns. Therefore, the rebuilding ratio is

$\begin{matrix} {{R\left( C^{\prime} \right)} = \frac{{s\left( {{{R(C)}{p\left( {k + 1} \right)}} - p} \right)} + {\left( {s - 1} \right)p} + p}{p\left( {{sk} + 1} \right)}} \\ {= {{R(C)}\frac{s\left( {k + 1} \right)}{{sk} + 1}}} \\ {= {{R(C)}\left( {1 + \frac{s - 1}{{sk} + 1}} \right)}} \end{matrix}$

and the proof is completed.

Theorem 10 gives us the ratio of the s-duplication of a code C as a function of its ratio R(C). As a result, for the optimal-ratio code in Theorem 8, the ratio of its duplication code is slightly more than 1/2, as the following corollary suggests.

Corollary 11

The s-duplication of the code in Theorem 8 has ratio

${\frac{1}{2}\left( {1 + \frac{s - 1}{{s\left( {m + 1} \right)} + 1}} \right)},$

which is

$\frac{1}{2} + \frac{1}{2\left( {m + 1} \right)}$

for large s.

For example, we can rebuild the column 1⁽⁰⁾ in FIG. 4 by accessing the elements in rows {0, 1} and in columns 0⁽⁰⁾, 2⁽⁰⁾, 0⁽¹⁾, 2⁽¹⁾, R, Z, and all the elements in column 1⁽¹⁾. The rebuilding ratio for this code is 4/7.

Using duplication we can have arbitrarily large number of columns, independent of the number of rows. Moreover the above corollary shows that it also has an almost optimal ratio.

Next we will show that if we restrict ourselves to codes constructed using Construction 1 and duplication, the code using the standard basis and duplication has optimal asymptotic rate.

In order to show that, we define a related graph. Define the directed graph D_(m)=D_(m)(V, E) as V={wεF₂ ^(m):w≠0}, and E={(w₁/w₂):|w₂\w₁|=1 mod 2}. Hence the vertices are the nonzero binary vectors of length m, and there is a directed edge from w₁ to w₂ if |w₂\w₁| is odd size. From any induced subgraph H of D_(m), we construct the code C(H) from the vertices of H using Construction 1. By Lemma 7 we know that a directed edge from w₁ to w₂ in H means f_(w) ₂ (X_(w) ₂ )∩f_(w) ₁ (X_(w) ₂ )=Ø, so only half of the information from the column corresponding to w₁ is accessed while rebuilding the column corresponding to w₂. For a directed graph D=D(V, E), let S and T be two disjoint subsets of its vertices. We define the density of the set S to be

$d_{S} = \frac{E_{S}}{|S|^{2}}$

and the density between S and T to be

${d_{S,T} = \frac{E_{S,T}}{\left. 2 \middle| \left. S||T \right. \right|}},$

where E_(S) is the number of edges with both of its endpoints in S, and E_(S,T) is the number of edges incident with a vertex in S and a vertex in T. The following theorem shows that the asymptotic ratio of any code constructed using Construction 1 and duplication is a function of the density of the corresponding graph H.

Theorem 12.

Let H be an induced subgraph of D_(m). Let C_(s)(H) be the s-duplication of the code constructed using the vertices of H and Construction 1. Then the asymptotic ratio of C_(s)(H) is

${\lim\limits_{s\rightarrow\infty}{R\left( {C_{s}(H)} \right)}} = {1 - \frac{d_{H}}{2}}$

Proof: Let the set of vertices and edges of H be V(H)={v_(i)} and E(H) respectively. Denote by v_(i) ^(l), v_(i)εV(H), lε[0, s−1], the l-th copy of the column corresponding to the vector v_(i). In the rebuilding of column v_(i) ^(l), lε[0, s−1], each remaining systematic column v_(j) ^(k), kε[0, s−1], needs to access all of its 2^(m) elements unless |v_(i)\v_(j)| is odd, and in that case it only has to access 2^(m−1) elements. Hence the total amount of accessed information for rebuilding this column is

(s|V(H)|−1)2^(m)−deg⁺(v _(i))s2^(m−1),

where deg⁺ is the indegree of v_(i) in the induced subgraph H. Averaging over all the columns in C_(s)(H) we get the ratio:

$\begin{matrix} {{R\left( {C_{s}(H)} \right)} = \frac{{\sum_{v_{i}^{l} \in {C_{s}{(H)}}}{\left( s \middle| {V(H)} \middle| {- 1} \right)2^{m}}} - {{\deg^{+}\left( v_{i} \right)}{s2}^{m - 1}}}{\left. s \middle| {V(H)} \middle| {\left( s \middle| {V(H)} \middle| {+ 1} \right)2^{m}} \right.}} \\ {= \frac{\left. s \middle| {V(H)} \middle| {{\left( s \middle| {V(H)} \middle| {- 1} \right)2^{m}} - {s^{2}{\sum_{v_{i} \in {V{(H)}}}{{\deg^{+}\left( v_{i} \right)}2^{m - 1}}}}} \right.}{\left. s \middle| {V(H)} \middle| {\left( s \middle| {V(H)} \middle| {+ 1} \right)2^{m}} \right.}} \\ {= \frac{\left. s \middle| {V(H)} \middle| {{\left( s \middle| {V(H)} \middle| {- 1} \right)2^{m}} - s^{2}} \middle| {E(H)} \middle| 2^{m - 1} \right.}{\left. s \middle| {V(H)} \middle| {\left( s \middle| {V(H)} \middle| {+ 1} \right)2^{m}} \right.}} \end{matrix}$ Hence ${\lim\limits_{s\rightarrow\infty}{R\left( {C_{s}(H)} \right)}} = {{1 - \frac{\left| {E(H)} \right|}{\left. 2 \middle| {V(H)} \right|^{2}}} = {1 - {\frac{d_{H}}{2}.}}}$

We conclude from Theorem 12 that the asymptotic ratio of any code using duplication and a set of binary vectors {v_(i)} is a function of the density of the corresponding induced subgraph of D_(m) with {v_(i)} as its vertices. Hence the induced subgraph of D_(m) with maximal density corresponds to the code with optimal asymptotic ratio. It is easy to check that the induced subgraph with its vertices as the standard basis {e_(i)}_(i=1) ^(m) has density

$\frac{m - 1}{m}.$

In fact this is the maximal possible density among all the induced subgraph as Theorem 14 suggests, but in order to show it we will need the following technical lemma.

Lemma 13.

Let D=D(V, E) be a directed graph and S, T be a partition of V, i.e., S∩T=Ø, S∩T=V, then

d _(v)≦max{d_(S) ,d _(T) ,d _(S,T)}

Proof: Note that

$d_{V} = {\frac{\left| S \middle| {}_{2}{d_{S} +} \middle| T \middle| {}_{2}{d_{T} + 2} \middle| \left. S||T \right. \middle| d_{S,T} \right.}{|V|^{2}}.}$

W.l.o.g assume that d_(S)≧d_(T) therefore if d_(S)≧D_(S,T),

$\begin{matrix} {d_{V} = \frac{\left| S \middle| {}_{2}{d_{S} +} \middle| T \middle| {}_{2}{d_{T} + 2} \middle| \left. S||T \right. \middle| d_{S,T} \right.}{|V|^{2}}} \\ {{\leqq \frac{\left| S \middle| {}_{2}{d_{S} +} \middle| T \middle| {}_{2}{d_{S} -} \middle| T \middle| {}_{2}{d_{S} +} \middle| T \middle| {}_{2}{d_{T} + 2} \middle| \left. S||T \right. \middle| d_{S} \right.}{|V|^{2}}}} \\ {{= \frac{\left. {{d_{S}\left( \left| S \middle| {+ |T|} \right. \right)}^{2} -} \middle| T \middle| {}_{2}\left( {d_{S} - d_{T}} \right) \right.}{|V|^{2}}}} \\ {{\leqq {d_{S}.}}} \end{matrix}$

If d_(S,T)≧max{d_(S), d_(T)} then,

$\begin{matrix} {d_{V} = \frac{\left| S \middle| {}_{2}{d_{S} +} \middle| T \middle| {}_{2}{d_{T} + 2} \middle| \left. S||T \right. \middle| d_{S,T} \right.}{|V|^{2}}} \\ {{\leqq \frac{\left| S \middle| {}_{2}{d_{S,T} +} \middle| T \middle| {}_{2}{d_{S,T} + 2} \middle| \left. S||T \right. \middle| d_{S,T} \right.}{|V|^{2}}}} \\ {{= d_{S,T}}} \end{matrix}$

and the result follows.

Now we are ready to prove the optimality of the duplication of the code using standard basis, if we assume that the number of copies s tends to infinity.

Theorem 14.

For any induced subgraph H of D_(m),

$d_{H} \leqq {\frac{m - 1}{m}.}$

So the optimal asymptotic ratio among all codes constructed using duplication and Construction 1 is

$\frac{1}{2}\left( {1 + \frac{1}{m}} \right)$

and is achieved using the standard basis.

Proof: We say that a binary vector is an even (odd) vector if it has an even (odd) weight. For two binary vectors w₁, w₂, |w₂\w₁| being odd is equivalent to

1=w ₂ · w ₁ =w ₂·((1, . . . , 1)+w ₁)=∥w ₂∥₁ +w ₂ ·w ₁.

Hence, one can check that when w₁, w₂ have the same parity, there are either no edges or two edges between them. Moreover, when their parities are different, there is exactly one edge between the two vertices.

When m=1, the graph D₁ has only one vertex and the only nonempty induced subgraph is itself. Thus,

$d_{H} = {d_{D_{1}} = {0 = {\frac{m - 1}{m}.}}}$

When m=2, the graph D₂ has three vertices and one can check that the induced subgraph with maximum density contains w₁=(1,0), w₂=(0, 1), and the density is 1/2=(m−1)/m.

For m>2, assume to the contrary that there exists a subgraph of D_(m) with density higher than

$\frac{m - 1}{m}.$

Let H be the smallest subgraph of D_(m) (with respect to the number of vertices) among the subgraphs of D_(m) with maximal density. Hence for any subset of vertices S ⊂V(H), we have d_(S)<d_(H). Therefore from Lemma 13 we conclude that for any partition S, T of V(H), d_(H)≦d_(S,T). If H contains both even and odd vectors, denote by S and T the set of even and odd vectors of H respectively. Since between any even and any odd vertex there is exactly one directed edge we get that

${d_{H} \leqq d_{S,T}} = {\frac{1}{2}.}$

However

${\frac{1}{2} < \frac{m - 1}{m} < d_{H}},$

and we get a contradiction. Thus H contains only odd vectors or even vectors.

Let V(H)={v₁, . . . , v_(k)}. If this set of vectors is independent then k≦m and the outgoing degree for each vertex v_(i) is at most k−1 hence

$d_{H} = {\frac{E(H)}{\left| {V(H)} \right|^{2}} \leqq \frac{k\left( {k - 1} \right)}{k^{2}} \leqq \frac{m - 1}{m}}$

and we get a contradiction. Hence assume that the dimension of the subspace spanned by these vectors in F₂ ^(m) is l<k where v₁, v₂, . . . v_(l) are basis for it. Define S={v₁, . . . v_(l)}, T={v_(l+1), . . . , v_(k)}. The following two cases show that the density cannot be higher than

$\frac{m - 1}{m}.$

H contains only odd vectors: Let uεT. Since uεspan{S} there is at least one vεS such that u·v≠0 and thus (u,v), (v,u)∉E(H), therefore the number of directed edges between u and S is at most 2(l−1) for all uεT, which means

${d_{H} \leqq d_{S,T} \leqq \frac{\left. {2\left( {l - 1} \right)} \middle| T \right|}{\left. 2 \middle| \left. S||T \right. \right|}} = {\frac{l - 1}{l} \leqq \frac{m - 1}{m}}$

and we get a contradiction.

H contains only even vectors: Since the v_(i)'s are even the dimension of span{S} is at most m−1 (since for example (1, 0, . . . , 0)∉span{S}) thus l≦m−1. Let H* be the induced subgraph of D_(m+1) with vertices V(H*)={(1, v_(i))|v_(i)εV(H))}. It is easy to see that all the vectors of H* are odd, ((1, v_(i)), (1, v_(j)))εE(H*) if and only if (v_(i), v_(j))εE(H), and the dimension of span{V(H*)} is at most l+1≦m. Having already proven the case for odd vectors, we conclude that

$\begin{matrix} {d_{H} = {d_{H^{*}} \leqq \frac{{\dim \left( {{span}\left\{ {V\left( H^{*} \right)} \right\}} \right)} - 1}{\dim \left( {{span}\left\{ {V\left( H^{*} \right)} \right\}} \right)}}} \\ {{\leqq \frac{l + 1 - 1}{l + 1}}} \\ {{{\leqq \frac{m - 1}{m}},}} \end{matrix}$

and we get a contradiction.

V. FINITE FIELD SIZE OF A CODE

In this section, we address the problem of finding proper coefficients in the parities in order to make the code an MDS. We have already shown that if a code is over some large enough finite field F, it can be made MDS (Theorem 3). In the following, we will discuss in more details on the field size required to make three kinds of codes MDS: (1) the optimal code in Theorem 8, (2) its duplication in Corollary 11, and (3) a modification of the code in Example 1. Note that both the codes in (2) and (3) have asymptotic optimal ratio.

Consider the (m+3, m+1) code C constructed by Theorem 8 and the vectors {e_(i)}_(i=0) ^(m). Let the information in row i, column j be a_(i,j)εF. Let its row coefficient be α_(i,j) and zigzag coefficient be β_(i,j). For a row set R_(u)={a_(u,0), a_(u,1), . . . , a_(u,m)}, the row parity is r_(u)=Σ_(j=0) ^(m)α_(u,j)α_(u,j). For a zigzag set z_(u)={a_(u,0), a_(u+e) _(l) _(,1), . . . , a_(u+e) _(m) _(,m)}, the zigzag parity is z_(u)=Σ_(j=0) ^(m)β_(u+e) _(j) _(,j)α_(u+e) _(j) _(,j).

Recall that the (m+3, m+1) code is MDS if we can recover the information from up to two columns erasures. It is clear that none of the coefficients α_(i,j), β_(i,j), can be zero. Moreover, if we assign all the coefficients as α_(i,j)=β_(i,j)=1 we get that in an erasure of two systematic columns the set of equations derived from the parity columns are linearly dependent and thus not solvable (the sum of the equations from the row parity and the sum of those from the zigzag parity will be the same). Therefore the coefficients need to be from a field with more than 1 nonzero element, thus a field of size at least three is necessary. The construction below surprisingly shows that in fact F₃ is sufficient.

Construction 2.

For the code C in Theorem 8 over F₃, define u_(j)=Σ_(l=0) ^(j)e_(l) for 0≦j≦m. Assign row coefficients as α_(i,j)=1 for all i, j, and zigzag coefficients as

β_(i,j)=2^(i·u),

where i=(i₁, . . . , I_(m)) is represented in binary and the calculation in the exponent is done over F₂.

Theorem 15.

Construction 2 is an (m+3, m+1) MDS code.

Proof: It is easy to see that if at least one of the two erased columns is a parity column then we can recover the information. Hence we only need to show that we can recover from any erasure of two systematic columns. In an erasure of two systematic columns i, jε[0, m], i<j, we access the entire remaining information in the array. For rε[0,2^(m)−1] set r′=r+e_(i)+e_(j), and recall that a_(r,i)εZ_(l) iff l=r+e_(i), thus a_(r,i), a_(r′,j)εZ_(r+e) _(i) and a_(r,j), a_(r′,i)εZ_(r+e) _(i). From the two parity columns we need to solve the following equations (for some y₁, y₂, y₃, y₄εF₃)

${\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \beta_{r,i} & 0 & 0 & \beta_{r^{\prime},j} \\ 0 & \beta_{r,j} & \beta_{r^{\prime},i} & 0 \end{bmatrix}\begin{bmatrix} a_{r,i} \\ a_{r,j} \\ a_{r^{\prime},i} \\ a_{r^{\prime},j} \end{bmatrix}} = \begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{bmatrix}$

This set of equations is solvable iff

β_(r,i)β_(r′i)≠β_(r,j)β_(r′,j).  (12)

Note that the multiplicative group of F₃\0 is isomorphic to the additive group of F₂, hence multiplying two elements in F₃\0 is equivalent to summing up their exponent in F₂ when they are represented as a power of the primitive element of the field F₃. For columns 0≦i<j≦m and rows r, r′ defined above, we have

β_(r,i)β_(r′,i)=2^(r·u) ^(i) ^(+r′·u) ^(i) =2^((r+r′)·u) ^(i) =2^((e) ^(i) ^(+e) ^(j) ^()·Σ) ^(l=0) ^(i) ^(e) ^(l) =2^(e) ^(i) ² =2.

However in the same manner we derive that

β_(r,j)β_(r′,j)=2^((r+r′)·u) ^(j) =2^((e) ^(i) ^(+e) ^(j) ^()·Σ) ^(l=0) ^(j) ^(e) ^(l) =2^(e) ^(i) ² ^(+e) ^(j) ² =2⁰=1.

Hence (12) is satisfied and the code is MDS.

Remark: The above proof shows that β_(r,i)≠β_(r′,i), and β_(r,j)=β_(r′,j) for i<j. And Equation (12) is a necessary and sufficient condition for a MDS code for vectors v_(i)≠v_(j). Moreover the coefficients in FIG. 3 are assigned by Construction 2.

Next we discuss the finite field size of the duplication of the optimal code (the code in Corollary 11). For the s-duplication code C′ in Corollary 11, denote the coefficients for the element in row i and column j^((t)) by α_(i,j) ^((t)) and β_(i,j) ^((t)), 0≦t≦s−1. Let F_(q) be a field of size q, and suppose its elements are {0, a⁰, a¹, . . . , a^(q−2)} for some primitive element a.

Construction 3.

For the s-duplication code C′ in Corollary 11 over F_(q), assign a_(i,j) ^((t))=1 for all i, j, t. For odd q, let s≦q−1 and assign for all tε[0, s−1]

$\beta_{i,j}^{(t)} = \left\{ \begin{matrix} {a^{t + 1},} & {{{if}\mspace{14mu} {u_{j} \cdot i}} = 1} \\ {a^{t},} & {o.w.} \end{matrix} \right.$

where u_(j)=Σ_(l=0) ^(j)e_(l). For even q (powers of 2), let s≦q−2 and assign for all tε[0, s−1]

$\beta_{i,j}^{(t)} = \left\{ \begin{matrix} {a^{{- t} - 1},} & {{{if}\mspace{14mu} {u_{j} \cdot i}} = 1} \\ {a^{t + 1},} & {o.w.} \end{matrix} \right.$

Notice that the coefficients in each duplication has the same pattern as Construction 2 except that values 1 and 2 are replaced by a^(t) and a^(t+1) if q is odd (or a^(t+1) and a^(−t−1) if q is even).

Theorem 16.

Construction 3 is an (s(m+1)+2, s(m+1)) MDS code.

Proof: For the two elements in columns i^((t) ¹ ⁾, i^((t2)) and row r, t₁≠t₂, we can see that they are in the same row set and the same zigzag set. The corresponding two equations from the two parities are linearly independent iff

β_(r,i) ^((t) ¹ ⁾≠β_(r,i) ^((t) ² ⁾,  (13)

which is satisfied by the construction.

For the four elements in columns i^((t) ¹ ⁾, j^((t) ² ⁾ and rows r, r′=r+e_(i)+e_(j), 0≦t₁, t₂≦s−1, 0≦i<j≦m, the code is MDS if

β_(r,i) ^((t) ¹ ⁾β_(r′,i) ^((t) ¹ ⁾≠β_(r,j) ^((t) ² ⁾β_(r′,j) ^((t) ² ⁾

by Equation (12). By the remark after Theorem 15, we know that β_(r,i) ^((t) ¹ ⁾≠β_(r′,i) ^((t) ¹ ⁾, and β_(r,j) ^((t) ² ⁾≠β_(r′,j) ^((t) ² ⁾=αa^(x) for some x. When q is odd,

β_(r,i) ^((t) ¹ ⁾β_(r′,i) ^((t) ¹ ⁾=α^(t) ¹ α^(t) ¹ ⁺¹α^(2t) ¹ ⁺¹≠α^(2x)

for any x and t₁. When q is even,

β_(r,i) ^((t) ¹ ⁾β_(r′,i) ^((t) ¹ ⁾=α^(t) ¹ ⁺¹α^(−t) ¹ ⁻¹=α⁰≠α^(2x)

for any t₁ and 1≦x≦q−2 (mod q−1). And by construction, x=t₂+1 or x=−t₂−1 for 0≦t₂≦s−1≦q−3, so 1≦x≦q−2 (mod q−1). Hence, the construction is MDS.

Remark: For two identical permutations f_(i) ^((t) ¹ ⁾=f_(i) ^((t) ² ⁾, Equation (13) is necessary and sufficient condition for an MDS code.

Theorem 17.

For an MDS s-duplication code, we need a finite field F_(q) of size q≧s+1. Therefore, Theorem 16 is optimal for odd q.

Proof: Consider the two information elements in row i and columns j^((t) ¹ ⁾, j^((t) ² ⁾, which are in the same row and zigzag sets, for t₁≠t₂ε[0, s−1]. The code is MDS only if

$\begin{bmatrix} \alpha_{i,j}^{t_{1}} & \alpha_{i,j}^{t_{2}} \\ \beta_{i,j}^{t_{1}} & \beta_{i,j}^{t_{2}} \end{bmatrix}\quad$

has full rank. All the coefficients are nonzero (consider erasing a parity column and a systematic column). Thus, (α_(i,j) ^(t) ¹ )⁻¹β_(i,j) ^((t) ¹ ⁾≠(α_(i,j) ^((t) ² ⁾)⁻¹β_(i,j) ^((t) ² ⁾, and (a_(i,j) ^((i)))⁻¹β_(i,j) ^((i)) are distinct nonzero elements in F_(q), for iε[0, s−1]. So q≧s+1.

For instance, the coefficients in FIG. 4 are assigned as Construction 3 and F₃ is used. One can check that any two column erasures can be rebuilt in this code.

Consider for example an s-duplication of the code in Theorem 8 with m=10, the array is of size 1024×(11s+2). For s=2 and s=6, the ratio is 0.522 and 0.537 by Corollary 11, the code length is 24 and 68, and the field size needed can be 4 and 8 by Theorem 16, respectively. Both of these two sets of parameters are suitable for practical applications.

As noted before the optimal construction yields a ratio of 1/2+1/m by using duplication of the code in Theorem 8. However the field size is a linear function of the number of duplications of the code. Is it possible to extend the number of columns in the code while using a constant field size? We know how to get O(m³) columns by using O(m²) duplications of the optimal code, however, the field size is O(m²). The following code construction has roughly the same parameters: O(m³) columns and an ratio of

${\frac{1}{2} + {O\left( \frac{1}{m} \right)}},$

however it requires only a constant field size of 9. Actually this construction is a modification of Example 1.

Construction 4.

Let 3|m, and consider the following set of vectors S

F₂ ^(m): for each vector v=(v₁, . . . , v_(m))εS, ∥v∥₁=3 and v_(i) ₁ , v_(i) ₂ , v_(i) ₃ =1 for some i₁ε[1, m/3], i₂ε[m/3+1, 2m/3], i₃ε[2m/3+1, m]. For simplicity, we write v={v₁, i₂, i₃}. Construct the (k+2, k) code as in Construction 1 using the set of vectors S, hence the number of systematic columns is

$k = {{S} = {\left( \frac{m}{3} \right)^{3} = {\frac{m^{3}}{27}.}}}$

For any iε[jm/3+1, (j+1)m/3] and some j=0, 1, 2, define a row vector M_(i)=Σ_(l=jm/3+1) ^(i)e_(l). Then define a m×3 matrix

M _(v) =[M _(i) ₁ ^(T) M _(i) ₂ ^(T) M _(i) ₃ ^(T)]

for v={i₁, i₂, i₃}. Let a be a primitive element of F₉. Assign the row coefficients as 1 and the zigzag coefficient for row r, column v as a^(t), where t=rM_(v)εF₂ ³ (in its binary expansion).

For example, let m=6, and v={1, 4, 6}=(1,0,0,1,0,1)εS. The corresponding matrix is

$M_{v} = {\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{bmatrix}^{T}.}$

For row r=26=(0,1,1,0,1,0), we have

t=rM _(v)=(0,1,1)=3,

and the zigzag coefficient is a³.

Theorem 18.

Construction 4 is a (k+2, k) MDS code with array size 2^(m)×(k+2) and k=m³/27. Moreover, the rebuilding ratio is

$\frac{1}{2} + \frac{9}{2m}$

for large m.

Proof: For each vector vεS, there are 3(m/3−1)² vectors uεS such that they have one “1” in the same location as v, i.e. |v\u|=2. Hence by Theorem 6 and Lemma 7, for large m the ratio is

${\frac{1}{2} + \frac{3\left( {\left( \frac{m}{3} \right) - 1} \right)^{2}}{2\left( {\frac{m^{3}}{27} + 1} \right)}} \approx {\frac{1}{2} + {\frac{9}{2m}.}}$

Next we show that the MDS property of the code holds. Consider columns u, v for some u={i₁, i₂, i₃}≠v={j₁, j₂, j₃} and i₁, j₁ε[1, m/3], i₂, j₂ε[m/3+1, 2m/3], i₃, j₃ε[2m/3+1, m]. Consider rows r and r′=r+u+v. The condition for the MDS property from Equation (12) becomes

a ^(rM) ^(u) ^(T) ^(+r′M) _(u) ^(T) ^(mod 8) ≠a ^(rM) ^(v) ^(T) ^(+r′M) ^(v) ^(T) ^(mod 8)  (14)

where each vector of length 3 is viewed as an integer in [0,7] and the addition is usual addition mod 8. Since v≠u, let lε[1,3] be the largest index such that i_(l)≠j_(l). W.l.o.g. assume that i_(l)<j_(l), hence by the remark after Theorem 15

rM _(i) _(l) ^(T) ≠r′M _(i) _(l) ^(T)  (15)

and

rM _(j) _(l) ^(T) ≠r′M _(j) _(l) ^(T).  (16)

Note that for all t, l<t≦3, i_(t)=j_(t), then since r′M_(i) _(l) ^(T)=(r+e_(i) _(t) +e_(j) _(i) )M_(i) _(t) ^(T)=rM_(i) _(t) ^(T), we have

rM _(i) _(t) ^(T) =r′M _(i) _(t) ^(T) =rM _(j) _(t) ^(T) =r′M _(j) _(t) ^(T).  (17)

It is easy to infer from Equations (15), (16), and (17) that the l-th bit in the binary expansions of rM_(u) ^(T)+r′M_(u) ^(T) mod 8 and rM_(v) ^(T)+r′M_(v) ^(T) mod 8 are not equal. Hence Equation (14) is satisfied, and the result follows.

Notice that if we do mod 15 in Equation (14) instead of mod 8, the proof still follows because 15 is greater than the largest possible sum in the equation. Therefore, a field of size 16 is also sufficient to construct an MDS code, and it is easier to implement in a storage system.

Construction 4 can be easily generalized to any constant c such that it contains O(m^(c)) columns and it uses the field of size at least 2^(c)+1. For simplicity assume that c|m, and simply construct the code using the set of vectors {v}⊂F₂ ^(m) such that ∥v∥₁=c, and for any jε[0, c−1], there is unique i_(j)ε[jm/c+1, (j+1)m/c] and v_(i) _(j) =1. Moreover, the finite field of size 2^(c+1) is also sufficient to make it an MDS code. When c is odd the code has ratio of

$\frac{1}{2} + \frac{c^{2}}{2m}$

for large m.

VI. DECODING OF THE CODES

In this section, we will discuss decoding algorithms of the proposed codes in case of column erasures as well as a column error. The algorithms work for both Construction 1 and its duplication code.

Let C be a (k+2, k) MDS array code defined by Construction 1 (and possibly duplication). The code has array size 2^(m)×(k+2). Let the zigzag permutations be f_(j), jε[0,k−1] which are not necessarily distinct. Let the information elements be a_(i,j), and the row and zigzag parity elements be r_(i) and z_(i), respectively, for iε[0,2^(m)−1], jε[0,k−1]. W.l.o.g. assume the row coefficients are a_(i,j)=1 for all i, j. And let the zigzag coefficients be β_(i,j) in some finite field F.

The following is a summary of the erasure decoding algorithms mentioned in the previous sections.

A. Algorithm 1 (Erasure Decoding)

One erasure.

-   -   (1) One parity node is erased. Rebuild the row parity by

$\begin{matrix} {{r_{i} = {\sum\limits_{j = 0}^{k - 1}a_{i,j}}},} & (18) \end{matrix}$

-   -    and rebuild the zigzag parity by

$\begin{matrix} {z_{i} = {\sum\limits_{j = 0}^{k - 1}{\beta_{{f_{j}^{- 1}{(i)}},j}{a_{{f_{j}^{- 1}{(i)}},j}.}}}} & (19) \end{matrix}$

-   -   (2) One information node j is erased. Rebuild the elements in         rows X_(j) (see Construction (1) by rows), and those in rows         X_(j) by zigzags.

Two erasures.

-   -   (1) Two parity nodes are erased. Rebuild by Equations (18)         and (19) above.     -   (2) One parity node and one information node is erased. If the         row parity node is erased, rebuild by zigzags; otherwise rebuild         by rows.     -   (3) Two information nodes j₁ and j₂ are erased.         -   If f_(j) ₁ =f_(j) ₂ , for any iε[0,2^(m)−1], compute

$\begin{matrix} {{x_{i} = {r_{i} - {\sum\limits_{{j \neq j_{1}},j_{2}}\; a_{i,j}}}}{y_{i} = {z_{f_{j\; 1}{(i)}} - {\sum\limits_{{j \neq j_{1}},j_{2}}\; {\beta_{{f_{j}^{- 1}{f_{j_{1}}{(i)}}},j}a_{{f_{j}^{- 1}{f_{j_{1}}{(i)}}},j}}}}}} & (20) \end{matrix}$

Solve α_(i,j) ₁ , α_(i,j) ₂ from the equations

${\begin{bmatrix} 1 & 1 \\ \beta_{i,j_{1}} & \beta_{i,j_{2}} \end{bmatrix}\begin{bmatrix} a_{i,j_{1}} \\ a_{i,j_{2}} \end{bmatrix}} = \begin{bmatrix} x_{i} \\ y_{i} \end{bmatrix}$

-   -   Else, for any iε[0,2^(m)−1], set i′=i+f_(j) ₁ (0)+f_(j) ₂ (0),         and compute x_(i), x_(i′), y_(i), y_(i′) according to Equation         (20). Then solve a_(i,j) ₁ , a_(i,j) ₂ , a_(i′,j) ₁ , a_(i′,j) ₂         from equations

${\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \beta_{i,j_{1}} & 0 & 0 & \beta_{i^{\prime},j_{2}} \\ 0 & \beta_{i,j_{2}} & \beta_{i^{\prime},j_{1}} & 0 \end{bmatrix}\begin{bmatrix} a_{i,j_{1}} \\ a_{i,j_{2}} \\ a_{i^{\prime},j_{1}} \\ a_{i^{\prime},j_{2}} \end{bmatrix}} = {\begin{bmatrix} x_{i} \\ x_{i^{\prime}} \\ y_{i} \\ y_{i^{\prime}} \end{bmatrix}.}$

In case of a column error, we first compute the syndrome, then locate the error position, and at last correct the error. Let x₀, x₁, . . . , x_(p−1)εF. Denote f⁻¹(x₀, x₁, . . . , x_(p−1))=(x_(f) ⁻¹ ₍₀₎, . . . , x_(f) ⁻¹ _((p−1))) for a permutation f on [0, p−1]. The detailed algorithm is as follows.

B. Algorithm 2 (Error Decoding)

Compute for all iε[0,2^(m)−1]:

$s_{i,0} = {{\sum\limits_{j = 0}^{k - 1}a_{i,j}} - r_{i}}$ $s_{i,1} = {{\sum\limits_{j = 0}^{k - 1}{\beta_{{f_{j}^{- 1}{(i)}},j}a_{{f_{j}^{- 1}{(i)}},j}}} - z_{i}}$

Let the syndrome be S₀=(s_(0,0), s_(1,0), . . . , s₂ _(m) _(−1,0)) and S₁=(s_(0,1), s_(1,1), . . . , s₂ _(m) _(−1,1)).

-   -   If S₀=0 and S₁=0, there is no error.     -   Else if one of S₀, S₁ is 0, there is an error in the parity.         Correct it by Equation (18) or Equation (19).     -   Else, find the error location. For j=0 to k−1:

Compute for all iε[0,2^(m)−1], x_(i,j)=β_(i,j)s_(i,0).

Let X_(j)=(x_(0,j), . . . , x₂ _(m) _(−1,j)) and Y_(j)=f_(j) ⁻¹(X_(j)).

If Y_(j)=S₁, subtract S₀ from column j. Stop.

If no such j is found, there is more than one error.

If there is only one error, the above algorithm is guaranteed to find the error location and correct it, since the code is MDS, as the following theorem states.

Theorem 19.

Algorithm 2 can correct one column error.

Proof: Notice that each zigzag permutation f_(j) is the inverse of itself by Construction 1, or f_(j)=f_(j) ⁻¹. Suppose there is error in column j, and the error is E=(e₀, e₁, . . . , e₂ _(m) ⁻¹). So the received column j is the sum of the original information and E. Thus the syndromes are s_(i,0)=e_(i) and

s _(i,1)=β_(f) _(j) _((i),j) e _(f) _(j) _((i)).

For column t, tε[0,k−1], we have x_(i,t)=β_(i,t)s_(i,0)=β_(i,t)e_(i). Write Y_(t)=(y_(0,t), . . . , y₂ _(m) _(−1,t)) and then

y _(i,t) =x _(f) _(t) _((i),t)=β_(f) _(t) _((i),t) e _(f) _(t) _((i)).

We will show the algorithm finds Y_(t)=S₁ iff t=j, and therefore subtracting S₀=E from column j will correct the error. When t=j, y_(i,t)=s_(i,l), for all iε[0,2^(m)−1], so Y_(j)=S₁. Now suppose there is t≠j such that Y_(t)=S₁. Since the error E is nonzero, there exists i such that e_(f) _(j) _((i))≠0. Consider the indices i and i′=f_(t)f_(j)(i). y_(i,t)=s_(i,l) yields

β_(f) _(t) _((i),t) e _(f) _(t) _((i))=β_(f) _(j) _((i),j) e _(f) _(j) _((i)).  (21)

Case 1:

When f_(t)=f_(j), set r=f_(t)(i)=f_(j)(i), then Equation (21) becomes β_(r,t)e_(r)=β_(r,j)e_(r) with e_(r)≠0. Hence β_(r,t)=β_(r,j) which contradicts Equation (13).

Case 2:

When f_(t)≠f_(j), since f_(t), f_(j) are commutative and are inverse of themselves, f_(t)(i′)=f_(t)f_(t)f_(j)=f_(j)(i) and f_(j)(i′)=f_(j)f_(t)f_(j)(i)=f_(t)(i). Therefore y_(i′,t)=s_(i′,l) yields

β_(f) _(j) _((i),t) e _(f) _(j) _((i))=β_(f) _(t) _((i),j) e _(f) _(t) _((i)).

The above two equations have nonzero solution (e_(f) _(j) _((i)), e_(f) _(t) _((i))) iff

β_(f) _(t) _((i),t)β_(f) _(j) _((i),t)=β_(f) _(j) _((i),j)β_(f) _(t) _((i),j)

which contradicts Equation (12) with r=f_(t)(i), r′=f_(j)(i). Hence the algorithm finds the unique erroneous column.

-   -   If the computations are done in parallel for all iε[0,2^(m)−1],         then Algorithm 2 can be done in time O(k). Moreover, since the         permutations f_(i)'s only change one bit of a number in         [0,2^(m)−1] in the optimal code in Theorem 8, the algorithm can         be easily implemented.

C. Correcting Node Erasures and Element Errors in Zigzag Code

In array codes for storage systems, data is arranged in a 2D array. Each column in the array is typically stored in a separate disk and is called a node, and each entry in the array is called an element. In the conventional error model, disk failures correspond to an erasure or an error of an entire node. Therefore, array codes are usually designed to correct such entire node failures. However, if we consider different applications, such as the case of flash memory as storage nodes, element error is also possible. In other words, we may encounter only a few errors in a column as well as entire node erasures. For an MDS array code with two parities, the minimum Hamming distance is 3, therefore, it is not possible to correct a node erasure and a node error at the same time. However, since a Zigzag code has very long column lengths, we ask ourselves: is it capable of correcting a node erasure and some element errors?

Given a (k+2, k) Zigzag code generated by distinct binary vectors T={v₀, v₁, . . . , v_(k−1)}, the following algorithm called Z1 (designated with “Z” because it particularly applies to Zigzag codes) corrects a node erasure and an element error. Here we assume that the erasure and error are in different columns, and there is only a single element error in the systematic part of the array. The code has two parities and 2^(m) rows, and the zigzag permutations are f_(j)=v_(j), jε[0, k−1]. The original array is denoted by (a_(i,j)), the erroneous array is (â_(i,j)). The row coefficients are all ones, and the zigzag coefficients are i, j. Let x₀, x₁, . . . , x_(p−1)εF. Denote f⁻¹(x₀, x₁, . . . , x_(p−1))=(x_(f−1(0)), x_(f−1(1)), . . . , x_(f−1(p−1))) for a permutation f on [0, p−1].

Algorithm Z1.

Suppose column t is erased, and there is at most one element error in the remaining array. Compute for all iε[0,2m−1] the syndromes:

${s_{i,0} = {\sum\limits_{j \neq t}\begin{matrix} {\hat{a}}_{i,j} & r_{i} \end{matrix}}},{s_{i,1} = {{\sum\limits_{j \neq t}\beta_{{f_{j}^{- 1}{(i)}},{j^{\hat{a}}{f_{j}^{- 1}{(i)}}},j}} - {z_{i}.}}}$

Let the syndrome be S₀=(s_(0,0), s_(1,0), . . . , s₂ _(m) _(−1,0)) and S₁×(s_(0,1), s_(1,1), . . . , s₂ _(m) _(−1,1)).

Compute for all iε[0,2^(m)−1], x_(i)=β_(i,t)s_(i,0). Let X=(x₀, . . . , x₂ _(m) ⁻¹), Y=f_(t) ⁻¹(S₁), W X=Y.

-   -   If W=0, there is no element error. Assign column t as −S₀.     -   Else, there will be two rows r, r′ such that w_(r), w_(r′) are         nonzero. Find j such that v_(j)=r+r′+v_(t). The error is in         column j.     -   If

${\frac{w_{r}}{w_{r^{\prime}}} = \frac{\beta_{r,t}}{\beta_{r,j}}},$

then the error is at row r, and assign

$a_{r,j} = {{\hat{a}}_{r,j}{\frac{W_{r}}{\beta_{r,t}}.}}$

-   -   Else if

${\frac{w_{r}}{w_{r^{\prime}}} = \frac{\beta_{r^{\prime},j}}{\beta_{r^{\prime},t}}},$

then the error is at row r′, and assign

$a_{r^{\prime},j} = {{\hat{a}}_{r^{\prime},j}{\frac{W_{r^{\prime}}}{\beta_{r^{\prime},t}}.}}$

-   -   Else there is more than one error.

Theorem Z2.

The above algorithm can correct a node erasure and a systematic element error.

Proof Suppose column t is erased and there is an error at column j and row r. Define r′=r+v_(t)+v_(j). Let â_(r,j)=a_(r,j)+e. It is easy to see that x_(i)=y_(i)=−β_(i,t)α_(i,t) except when i=r,r′. Since the set of binary vectors {v₀, v₁, . . . , v_(k−1)} are distinct, we know that the error is in column j. Moreover, we have

x _(r)=+β_(r,t)α_(r,t)β_(r,t) e,

y _(r)=−β_(r,t)α_(r,t),

x _(r′)=−β_(r′,t)α_(r′,t),

y _(r′)=+β_(r′,t)β_(r,j) e.

Therefore, the difference between X and Y is

w _(r) =x _(r) −y _(r)=β_(r,t) e,

w _(r′) =−x _(r′) −y _(r′)=β_(r,j) e.

And we can see that no matter what e is, we always have

$\frac{w_{r}}{w_{r^{\prime}}} = {\frac{\beta_{r,t}}{\beta_{r^{\prime},j}}.}$

Similarly, if the error is at row r′, we will get

$\frac{w_{r}}{w_{r^{\prime}}} = {\frac{\beta_{r^{\prime},j}}{\beta_{r^{\prime},t}}.}$

By the MDS property of the code, we know that β_(r,t)β_(r′,t)≠β_(r,j)β_(r′,j) (see the remark after the proof of the finite field size 3). Therefore, we can distinguish between the two cases of an error in row r and in row r′.

Example Z3

Consider the code in FIG. 1. Suppose all of Column 0 is erased. And suppose there is an error in the 0-th element in Column 1. Namely, the erroneous symbol we read is {circumflex over (b)}₀=b₀ e- for some error e≠0εΦ₃, see FIG. 2. We can simply compute the syndrome, locate this error, and recover the original array. Since the erased column corresponds to the zero vector, and all the coefficients in column 0 are ones. The algorithm is simplified. For iε[0, 3], we compute the syndromes and subtract them, we get zeros in all places except row 0 and 2, which satisfy 0+2=(0,0)+(1,0)=(1,0)=e₁. Therefore, we know the location of the error is in column 1 and row 0 or 2. But since W₀=−W₂, we know the error is in {circumflex over (b)}₀ (If W₀=W₂, the error is in {circumflex over (b)}₂).

In practice, when we are confident that there are no element errors besides the node erasure, we can use the optimal rebuilding algorithm and access only half of the array to rebuild the failed node. However, we can also try to rebuild this node by accessing the other half of the array. Thus we will have two recovered version for the same node. If they are equal to each other, there are no element errors; if not, there are element errors. Thus, we have the flexibility of achieving optimal rebuilding ratio or correcting extra errors.

When there is one node erasure and more than one element errors in column j and row R={r₁, r₂, . . . , r_(l)}, following the same techniques, it is easy to see that the code is able to correct systematic errors if R∪(R+v_(j))≠R′∪(R′+v_(i)) for any set of rows R′ and any other column index i, and r_(i)≠r_(t)+v_(j) for any i, tε[l].

When the code has more than two parities, the Zigzag code can again correct element errors exceeding the bound by the Hamming distance. To detect errors, one can either compute the syndromes, or rebuild the erasures multiple times by accessing different e/r parts of the array.

Finally, it should be noted that if a node erasure and a single error happen in a parity column, then we cannot correct this error in the (k+2, k) code.

VII. GENERALIZATION OF THE CODE CONSTRUCTION

In this section we generalize Construction 1 into arbitrary number of parity nodes. Let n−k=r be the number of parity nodes. We will construct an (n,k) MDS array code, i.e., it can recover from up to r node erasures for arbitrary integers n, k. We will show this code has optimal rebuilding ratio of 1/r when a systematic node is erased. We assume that each systematic nodes stores M/k of the information and corresponds to columns [0,k−1]. The i-th parity node is stored in column k+i, 0≦i≦r−1, and is associated with zigzag sets {Z_(j) ^(i):jε[0, p−1]} where p is the number of rows in the array.

Construction 5.

Let the information array be A=(a_(i,j)) with size r^(m)×k for some integers k, m. Let T={v₀, . . . , v_(k−1)}

Z_(r) ^(m) be a subset of vectors of size k, where for each v=(v₁, . . . , v_(m))εT,

gcd(v ₁ , . . . , v _(m) ,r).  (22)

where gcd is the greatest common divisor. For any l, 0≦l≦r−1, and vεT we define the permutation f_(v) ^(l):[0, r^(m)−1]→[0, r^(m)−1] by f_(v) ^(l)(x)=x+lv, where by notation we use xε[0, r^(m)−1] both to represent the integer and its r-ary representation, and all the calculations are done over Z_(r). For example, for m=2, r=3, x=4, l=2, v=(0, 1),

f _((0,1)) ²(4)=4+2(0,1)=(1,1)+(0,2)=(1,0)=3,

One can check that the permutation f_(v) ^(l) in a vector notation is [2,0,1,5,3,4,8,6,7]. For simplicity denote the permutation f_(v) _(j) ^(l) as f_(j) ^(l) for v_(j)εT. For tε[0, r^(m)−1], we define the zigzag set Z_(t) ^(l) in parity node l as the elements a_(i,j) such that their coordinates satisfy f_(j) ^(l)(i)=t. In a rebuilding of systematic node i the elements in rows X_(i) ^(l)={xε[0, r^(m)−1]:x·v_(i)=r−l} are rebuilt by parity node l, for lε[0, r−1]. From Equation (22) we get that for any i and l, |X_(i) ^(l)|=r^(m−1).

Note that similar to Theorem 3, using a large enough field, the parity nodes described above form an (n,k) MDS array code under appropriate selection of coefficients in the linear combinations of the zigzags.

Consider the rebuilding of systematic node iε[0,k−1]. In a systematic column j≠i we need to access all the elements that are contained in the sets that belong to the rebuilding set of column i. Namely, in column j we need to access the elements in rows

∪_(l=0) ^(r−1) f _(j) ^(−l) f _(i) ^(l)(X _(i) ^(l)).  (23).

Equation (23) follows since the zigzags Z_(t) ^(l) for any tεf_(i) ^(l)(X_(i) ^(l)) are used to rebuild the elements of column i in rows X_(i) ^(l). Moreover the element in column j and zigzag Z_(t) ^(l) is a_(f) _(t) _(−l) _((t),j). The following lemma will help us to calculate the size of Equation (23), and in particular calculating the ratio of codes constructed by Construction 5.

Lemma 20.

For any v=(v₁, . . . , v_(m)), uεZ_(r) ^(m), and lε[0, r−1] such that gcd(v₁, . . . , v_(m), r)=1, define c_(v,u)=v·(v−u)−1. Then

${{{f_{u}^{- i}{f_{v}^{i}\left( X_{v}^{i} \right)}}\bigcap{f_{u}^{- j}{f_{v}^{j}\left( X_{v}^{j} \right)}}}} = \left\{ \begin{matrix} {{X_{v}^{0}},} & {{\left( {i - j} \right)c_{v,u}} = 0} \\ {0,} & {o.w.} \end{matrix} \right.$

In particular for j=0 we get

${{{f_{u}^{- i}{f_{v}^{i}\left( X_{v}^{l} \right)}}\bigcap X_{v}^{0}}} = \left\{ \begin{matrix} {{X_{v}^{0}},} & {{{if}\mspace{14mu} {lc}_{v,u}} = 0} \\ {0,} & {o.w.} \end{matrix} \right.$

Proof: Consider the group (Z_(r) ^(m), +). Note that X_(v) ⁰={x:x·v=0} is a subgroup of Z_(r) ^(m) and X_(v) ^(i)={x:x·v=r−i} is its coset. Therefore, X_(v) ^(i)=X_(v) ⁰+a_(v) ^(i), X_(v) ^(j)=X_(v) ⁰+a_(v) ^(j), for some a_(v) ^(i)εX_(v) ^(i), a_(v) ^(j)εX_(v) ^(j). Hence f_(u) ^(−i)f_(v) ^(i)(X_(v) ^(i))=X_(v) ⁰+a_(v) ^(i)+i(v−u) and f_(u) ^(−j)f_(v) ^(j)(X_(v) ^(j))=X_(v) ⁰+a_(v) ^(j)+j(v−u) are cosets of X_(v) ⁰. So they are either identical or disjoint. Moreover they are identical if and only if

a _(v) ^(i) −a _(v) ^(j)+(i−j)(v−u)εX _(v) ⁰,

i.e., (a_(v) ^(i)−a_(v) ^(j)+(i−j)(v−u))·v=0. But by definition of X_(v) ^(i) and X_(v) ^(j), a_(v) ^(i)·v=−i, a_(v) ^(j)·v=−j, so (i−j)·c_(v,u)=0 and the result follows.

The following theorem gives the ratio for any code of Construction 5.

Theorem 21.

The ratio for the code constructed by Construction 5 and set of vectors T is

$\frac{{\sum\limits_{v \in T}^{\;}{\sum\limits_{{u \neq v} \in T}^{\;}\frac{1}{\gcd \left( {r,c_{v,u}} \right)}}} + {T}}{{T}\left( {{T} - 1 + r} \right)},$

Which also equal to

$\frac{1}{r} + {\frac{\sum\limits_{v \in T}^{\;}{\sum\limits_{{u \in T},{u \neq v}}^{\;}{{{F_{u,v}\left( \overset{\_}{X_{v}^{0}} \right)}\bigcap\overset{\_}{X_{v}^{0}}}}}}{{T}\left( {{T} - 1 + r} \right)r^{m}}.}$

Here we define the function F_(u,v)(t)=f_(u) ^(−i)f_(v) ^(i)(t) for tεX_(v) ^(i).

Proof: By (23) and noticing that we access r^(m−1) elements in each parity node, the ratio is

$\begin{matrix} {\frac{{\sum\limits_{v \in T}^{\;}{\sum\limits_{{u \neq v} \in T}^{\;}{{\bigcup_{i = 0}^{r - 1}{f_{u}^{- i}{f_{v}^{i}\left( X_{v}^{i} \right)}x}}}}} + {{T}r^{m}}}{{T}\left( {{T} - 1 + r} \right)r^{m}}.} & (24) \end{matrix}$

From Lemma 20, and noticing that |{i:ic_(v,u)=0 mod r}|=gcd(r, c^(v,u)), we get

|∪_(i=0) ^(r−1) f _(u) ^(−i) f _(v) ^(i)(X _(v) ^(i))|=r ^(m−1) ×r/gcd(r,c _(v,u)).

And the first part follows. For the second part,

$\begin{matrix} \begin{matrix} {\frac{{\sum\limits_{v \in T}^{\;}{\sum\limits_{{u \neq v} \in T}^{\;}{{\bigcup_{i = 0}^{r - 1}{f_{u}^{- i}{f_{v}^{i}\left( X_{v}^{i} \right)}}}}}} + {{T}r^{m}}}{{T}\left( {{T} - 1 + r} \right)r^{m}} = \frac{\begin{matrix} {{\sum\limits_{v \in T}^{\;}{\sum\limits_{{u \neq v} \in T}^{\;}{X_{v}^{0}}}} +} \\ \begin{matrix} {{{\bigcup_{i = 0}^{r - 1}{f_{u}^{- i}{{f_{v}^{i}\left( X_{v}^{i} \right)}\backslash X_{v}^{0}}}}} +} \\ {{T}r^{m}} \end{matrix} \end{matrix}}{{T}\left( {{T} - 1 + r} \right)r^{m}}} \\ {= {\frac{1}{r} + \frac{\begin{matrix} {\sum\limits_{v \in T}^{\;}\sum\limits_{{u \neq v} \in T}^{\;}} \\ {{{\bigcup_{i = 0}^{r - 1}{f_{u}^{- i}{f_{v}^{i}\left( X_{v}^{i} \right)}}}\bigcap\overset{\_}{X_{v}^{0}}}} \end{matrix}}{{T}\left( {{T} - 1 + r} \right)r^{m}}}} \\ {= {\frac{1}{r} + {\frac{\begin{matrix} {\sum\limits_{v \in T}^{\;}\sum\limits_{{u \in T},{u \neq v}}^{\;}} \\ {{{F_{u,v}\left( \overset{\_}{X_{v}^{0}} \right)}\bigcap\overset{\_}{X_{v}^{0}}}} \end{matrix}}{{T}\left( {{T} - 1 + r} \right)r^{m}}.}}} \end{matrix} & (25) \end{matrix}$

The proof is completed.

Notice that X_(v) ⁰ represents elements not accessed for parity 0 (row parity), and F_(u,v)( X_(v) ⁰ ) are elements accessed for parity 1, 2, . . . , r−1. Therefore F_(u,v)( X_(v) ⁰ )∩ X_(v) ⁰ are the elements accessed excluding those for the row parity. In order to get a low rebuilding ratio, we need to minimize the second term in (25). We say that a family of permutation set {{f₀ ^(l)}_(l=0) ^(r−1), . . . , {f_(k−1) ^(l)}_(l=0) ^(r−1)} together with sets {{X₀ ^(l)}_(l=0) ^(r−1), . . . , {X_(k−1) ^(l)}_(l=0) ^(r−1)} is a family of orthogonal permutations if for any i, jε[0,k−1] the set {X_(i) ^(l}) _(i=0) ^(r−1) is an equally-sized partition of [0, r^(m−1)] and

$\frac{{{F_{j,i}\left( \overset{\_}{X_{i}^{0}} \right)}\bigcap\overset{\_}{X_{i}^{0}}}}{r^{m - 1}\left( {r - 1} \right)} = {\delta_{i,j}.}$

One can check that for r=2 the definition coincides with the previous definition of orthogonal permutations. It can be shown that the above definition is equivalent to that for any 0≦i≠j≦k−1, 0≦l≦r−1,

f _(j) ^(l)(X _(i) ⁰)=f _(i) ^(l)(X _(i) ^(l)).  (26)

Theorem 22.

The set {{f₀ ^(l)}_(l=0) ^(r−1), . . . , {f_(m) ^(l)}_(l=0) ^(r−1)} together with set {{X₀ ^(l)}_(l=0) ^(r−1), . . . , {X_(m) ^(l)}_(l=0) ^(r−1)} constructed by the vectors {e_(i)}_(i=0) ^(m) and Construction 5, where X₀ ^(l) is modified to be X₀ ^(l)={xεZ_(r) ^(m):x·(1, 1, . . . , 1)} for any lε[0, r−1] is a family of orthogonal permutations. Moreover the corresponding (m+1+r, m+1) code has optimal ratio of 1/r.

Proof: For 1≦i≠j≦m, c_(i,j)=e_(i)·(e_(i)−e_(j))−1=0, hence by Lemma 20 for any lε[0, r−1]

f _(j) ^(−l) f _(i) ^(l)(X _(i) ^(l))∩X _(i) ⁰ =X _(i) ⁰,

and Equation (26) is satisfied. For 1≦i≦m, and all 0≦l≦r−1,

$\begin{matrix} {{f_{0}^{- l}{f_{i}^{l}\left( X_{i}^{l} \right)}} = {f_{i}^{l}\left( \left\{ {{v:v_{i}} = {- l}} \right\} \right)}} \\ {= \left\{ {{v + {{le}_{i}:v_{i}}} = {- l}} \right\}} \\ {= {\left\{ {{v:v_{i}} = 0} \right\} = X_{i}^{0}}} \end{matrix}$

Therefore, f₀ ^(−l)f_(i) ^(l)(X_(i) ^(l))∩X_(i) ⁰=X_(i) ⁰, and Equation (26) is satisfied. Similarly,

$\begin{matrix} {{f_{i}^{- l}{f_{0}^{l}\left( X_{0}^{l} \right)}} = {f_{i}^{- l}\left( \left\{ {{v:{v \cdot \left( {1,\ldots \;,1} \right)}} = l} \right\} \right)}} \\ {= \left\{ {{v - {{le}_{i}:{v \cdot \left( {1,\ldots \;,1} \right)}}} = l} \right\}} \\ {= {\left\{ {{v:{v \cdot \left( {1,\ldots \;,1} \right)}} = 0} \right\} = {X_{0}^{0}.}}} \end{matrix}$

Hence again (26) is satisfied and this is a family of orthogonal permutations. By (25) we get that the ratio is 1/r, which is optimal according to Equation (1), and the result follows.

Surprisingly, one can infer from the above theorem that changing the number of parities from 2 to 3 adds only one node to the system, but reduces the ratio from 1/2 to 1/3 in the rebuilding of any systematic column.

The example in FIG. 7 shows a code with three systematic nodes and three parity nodes constructed by Theorem 22 with m=2. The code has an optimal ratio of 1/3. For instance, if column C₁ is erased, then accessing rows {0, 1, 2} in the remaining nodes will be sufficient for rebuilding. In FIG. 7, the first parity column C₃ corresponds to the row sums, and the corresponding identity permutations are omitted. The second and third parity columns C₄, C₅ are generated by the permutations f_(i) ¹, f_(i) ² respectively, for i=0, 1, 2. The elements are from F₇, where 3 is a primitive element.

Similar to the two-parity case, the following theorem shows that Theorem 22 achieves the optimal number of columns. In other words, the number of rows has to be exponential in the number of columns in any systematic MDS code with optimal ratio, optimal update, and r parities. This follows since any such optimal code is constructed from a family of orthogonal permutations.

Theorem 23.

Let {{f₀ ^(l)}_(l=0) ^(r−1), . . . , {f_(k−1) ^(l)}_(l=0) ^(r−1)} be a family of orthogonal permutations over the integers [0, r^(m)-1] together with the sets {{X₀ ^(l)}_(l=0) ^(r−1), . . . , {X_(k−1) ^(l)}_(l=0) ^(r−1)}, then k≦m+1.

Proof: We prove it by induction on m. When m=0, it is trivial that k≦1. Recall that orthogonality is equivalent to that any 0≦i≠j≦k−1, 0≦l≦r−1,

f _(j) ^(l)(X _(i) ⁰)=f _(i) ^(l)(X _(i) ^(l)).  (27)

Notice that for any permutations g, h₀ . . . , h_(r−1), {{h_(l)f₀ ^(l)g}_(l=0) ^(r−1), . . . , {h_(l)f_(k−1) ^(l)g}_(l=0) ^(r−1)}} are still a family of orthogonal permutations with sets {{g⁻¹(X₀ ^(l))}, . . . , {g⁻¹(X_(k−1) ^(l))}}. This is because

$\begin{matrix} {{h_{l}f_{j}^{l}{g\left( {g^{- 1}\left( X_{i}^{0} \right)} \right)}} = {h_{l}{f_{j}^{l}\left( X_{i}^{0} \right)}}} \\ {= {h_{l}{f_{i}^{l}\left( X_{i}^{l} \right)}}} \\ {= {h_{l}f_{i}^{l}{{g\left( {g^{- 1}\left( X_{i}^{l} \right)} \right)}.}}} \end{matrix}$

Therefore, w.l.o.g. we can assume X₀ ^(l)=[lr^(m−1), (l+1)r^(m−1)−1], and f₀ ^(l) is the identity permutation, for 0≦l≦r−1.

Let 1≦i≠j≦k−1, lε[0, r−1] and define

A=f _(j) ^(l)(X _(i) ⁰)=f _(i) ^(l)(X _(i) ^(l)),

B=f _(j) ^(l)(X _(i) ⁰ ∩X ₀ ⁰),

C=f _(i) ^(l)(X _(i) ^(l) ∩X ₀ ⁰).

Therefore B, C are subsets of A, and their compliments in A are

A\B=f _(j) ^(l)(X _(i) ⁰∩ X ₀ ⁰ ),

A\C=f _(i) ^(l)(X _(i) ^(l)∩ X ₀ ⁰ ).

From Equation (27) for any j≠0,

f _(j) ^(l)(X ₀ ⁰)=f ₀ ^(l)(X ₀ ^(l))=X ₀ ^(l)  (28)

hence,

B,C

X ₀ ^(l)  (29)

Similarly, for any j≠0, f_(j) ^(l)( X₀ ⁰ )= f_(j) ^(l)(X₀ ⁰)= X₀ ^(l) , hence

A\B,A\C

X ₀ ^(l) .  (30)

From Equations (29), (30) we conclude that B=C=A∩X₀ ¹, i.e.,

f _(j) ^(l)(X _(i) ⁰ ∩X ₀ ⁰)=f _(i) ^(l)(X _(i) ^(l) ∩X ₀ ⁰).  (31)

For each lε[0, r−1], jε[1, k−1] define {circumflex over (f)}_(j) ^(l)(x)=f_(j) ^(l)(x)−lr^(m−1) and {circumflex over (X)}_(j) ^(l)=X_(j) ^(l)∩X₀ ⁰ then,

$\begin{matrix} \begin{matrix} {{{\hat{f}}_{j}^{l}\left( \left\lbrack {0,{r^{m - 1} - 1}} \right\rbrack \right)} = {{f_{j}^{l}\left( X_{0}^{0} \right)} - {lr}^{m - 1}}} \\ {= {X_{0}^{l} - {lr}^{m - 1}}} \\ {{= \left\lbrack {0,{r^{m - 1} - 1}} \right\rbrack},} \end{matrix} & (32) \end{matrix}$

where Equation (32) follows from Equation (28). Moreover, since f_(i) ^(l) is bijective we conclude that {circumflex over (f)}_(i) ^(l) is a permutation on [0, r^(m−1)−1].

$\begin{matrix} \begin{matrix} {{{\hat{f}}_{i}^{l}\left( {\hat{X}}_{i}^{l} \right)} = {{f_{i}^{l}\left( {X_{i}^{l}\bigcap X_{0}^{0}} \right)} - {lr}^{m - 1}}} \\ {= {{f_{j}^{l}\left( {X_{i}^{0}\bigcap X_{0}^{0}} \right)} - {lr}^{m - 1}}} \\ {{= {{\hat{f}}_{j}^{l}\left( X_{i}^{0} \right)}},} \end{matrix} & (33) \end{matrix}$

where Equation (33) follows from Equation (31). Since {X_(i) ^(l)}_(l=0) ^(r−1) is a partition of [0, r^(m)-1], then {{circumflex over (X)}_(i) ^(l)}_(l=0) ^(r−1) is also a partition of X₀ ⁰=[0, r^(m−1)−1]. Moreover, since {circumflex over (f)}_(i) ^(l)({circumflex over (X)}_(i) ^(l))={circumflex over (f)}_(j) ^(l)({circumflex over (X)}_(i) ⁰) for any lε[0, r−1], and {circumflex over (f)}_(i) ^(l) is a bijection, we conclude

|{circumflex over (X)} _(i) ^(l) |=|{circumflex over (X)} _(i) ⁰|

for all lε[0, r−1], i.e., {{circumflex over (x)}_(i) ^(l)}, lε[0, r−1], is an equally sized partition of [0, r^(m−1)−1]. Therefore {{{circumflex over (f)}₁ ^(l)}_(l=0) ^(r−1), . . . , {f_(k−1) ^({circumflex over (l)})}_(l=0) ^(r−1)} together with {{{circumflex over (X)}₁ ^(l)}_(l=0) ^(r−1), . . . , {X_(k−1) ^({circumflex over (l)})}_(l=0) ^(r−1)} is a family of orthogonal permutations, hence by induction k−1≦m and the result follows.

After presenting the Construction of a code with optimal ratio of 1/r, we move on to deal with the problem of assigning the proper coefficient in order to satisfy the MDS property. This task turns out to be not easy when the number of parities r>2. The next theorem gives a proper assignment for the code with r=3 parities, constructed by the optimal construction given before. This assignment gives an upper bound on the required field size.

Theorem 24.

A field of size at most 2(m+1) is sufficient to make the code constructed by Theorem 22 with r=3 parities, a (m+4, m+1) MDS code.

Proof: Let F_(q) be a field of size q≧2(m+1). For any lε[0, m] let A_(l)=(a_(i,j)) be the representation of the permutation f_(e) _(l) ¹, by a permutation matrix with a slight modification and is defined as follows,

$a_{i,j} = \left\{ \begin{matrix} \alpha^{l} & {{f_{e_{l}}^{1}(j)} = {{i\mspace{14mu} {and}\mspace{14mu} {j \cdot e_{t}}} = 0}} \\ 1 & {{f_{e_{l}}^{1}(j)} = {{i\mspace{14mu} {and}\mspace{14mu} {j \cdot e_{t}}} \neq 0}} \\ 0 & {{otherwise},} \end{matrix} \right.$

where α is a primitive element of F_(q). Let W be the matrix that create the parities nodes, defined as

$W = {\begin{bmatrix} B_{0}^{0} & B_{1}^{0} & \ldots & B_{m}^{0} \\ B_{0}^{1} & B_{1}^{1} & \ldots & B_{m}^{1} \\ B_{0}^{2} & B_{1}^{2} & \ldots & B_{m}^{2} \end{bmatrix}.}$

Where B_(l) ^(j)=(A_(l))^(j) for lε[0, m] and jε[0,2]. It easy to see that indeed block row iε[0,2] in the block matrix m corresponds to parity i. We will show that this coefficient assignment satisfy the MDS property of the code. First we will show that under this assignment of coefficients the matrices A_(l) commute, i.e. for any l₁≠l₂ε[0, m], A_(l) ₁ A_(l) ₂ =A_(l) ₂ A_(l) ₁ . For simplicity, write

f_(e_(l₁))¹ = f₁, f_(e_(l₂))¹ = f₂, A_(l₁) = (a_(i, j)), A_(l₂) = (b_(i, j)), 3^(m) = p.

For a vector x=(x₀, . . . , x_(p−1)) and y=xA_(l), its j-th entry satisfies y_(j)=a_(f) ₁ _((j),j)x_(f) ₁ _((j)) for all jε[0, p−1]. And by similar calculation, z=xA_(l) ₁ A_(l) ₂ =yA_(l) ₂ will satisfy

z _(j) =b _(f) ₂ _((j),j) y _(f) ₂ _((j)) =b _(f) ₂ _((j),j) a _(f) ₁ _((f) ₂ _((j)),f) ₂ _((j)) x _(f) ₁ _((f) ₂ _((j))).

Similarly, if w=xA_(l) ₂ A_(l) ₁ , then

w _(j) =a _(f) ₁ _((j),j) b _(f) ₂ _((f) ₁ _((j)),f) ₁ _((j)) x _(f) ₂ _((f) ₁ _((j))).

Notice that

f ₁(j)·e _(l) ₂ =(j+e _(l) ₁ )e _(l) ₂ =j·e _(l) ₂ ,

so b_(f) ₂ _((j),j)=b_(f) ₂ _((f) ₁ _((j)),f) ₁ _((j)). Similarly, a_(f) ₁ _((j),j)=a_(f) ₁ _((f) ₂ _((j)),f) ₂ _((j)). Moreover,

f ₁(f ₂(j))=f ₂(f ₁(j))=j+e _(l) ₁ +e _(l) ₂ .

Hence, z_(j)=w₁ for all j and

xA _(l) ₁ A _(l) ₂ =z=w=xA _(l) ₂ A _(l) ₁

for all xεF₃ ^(m). Thus A_(l) ₁ A_(l) ₂ =A_(l) ₂ A_(l) ₁ .

Next we show for any i, A_(i) ³=α^(i)I. For any vector x, Let y=xA_(i) ³. Then

y _(j) =a _(f) ₁ _((j),j) a _(f) _(i) ₂ _((j),f) _(i) _((j)) a _(f) _(i) ₃ _((j),f) _(i) ₂ _((j)) x _(f) _(i) ₃ _((j)).

However, f_(i) ³(j)=j+3e_(i)=j (since the addition is done over F₃ ^(m)), and exactly one of j·e_(i), f_(i)(j)·e_(i), f_(i) ²(j)·e_(i) equals to 0. Thus y_(j)=a^(i)x_(j) or xA_(i) ³=a^(i)x for any x. Hence A_(i) ³=a^(i)I.

The code is MDS if it can recover from loss of any three nodes. With this assignment of coefficients the code is MDS iff any block sub matrices of sizes 1×1, 2×2, 3×3 of the matrix M are invertible. Let 0≦i<j<k≦m we will see that the matrix

$\quad\begin{bmatrix} I & I & I \\ A_{i} & A_{j} & A_{k} \\ A_{i}^{2} & A_{j}^{2} & A_{k}^{2} \end{bmatrix}$

is invertible. By Theorem 1 in [J. R. Silvester, The Mathematical Gazette, 84:460-467 (November 2000)] and the fact that all the blocks in the matrix commute we get that the determinant of this matrix equals to det(A_(k)−A_(j))·det(A_(k)−A_(i))·det(A_(j)−A_(i)). Hence we need to show that for any i>j, det(A_(i)−A_(j))≠0, which is equivalent to det(A_(i)A_(j) ^(−1−I)≠)0. Note that for any i, A_(i) ³=a^(i)I. Denote by A=A_(i)A_(j) ⁻¹, hence A³=(A_(i)A_(j) ⁻¹)³=A_(i) ³A_(j) ⁻³=a^(i−j)I≠I. Therefore

0≠det(A ³ −I)=det(A−I)det(A ² +A+I).

Therefore det(A−I)=det(A_(i)A_(j) ⁻¹−I)≠0. For a submatrix of size 2×2, we need to check that for i>j

${\det \left( \begin{bmatrix} I & I \\ A_{j}^{2} & A_{i}^{2} \end{bmatrix} \right)} = {{{\det \left( A_{j}^{2} \right)}{\det \left( {{A_{i}^{2}A_{j}^{- 2}} - I} \right)}} \neq 0.}$

Note that A⁶=(A_(i)A_(j) ⁻¹)⁶=a^(2(i−j))I≠I since

$0 < {i - j} \leq m < {\frac{q - 1}{2}.}$

Hence

0≠det(A ⁶ −I)=det(A ² −I)(A ⁴ +A ² +I),

and det(A²−I)=det(A_(i) ²A_(j) ⁻²−I)≠0 which concludes the proof.

For example, the coefficients of the parities in FIG. 7 are assigned as the above proof. Since m=2, the field of size 7 is sufficient. The primitive element is chosen to be 3. One can check that when losing any three columns we can still rebuild them.

VIII. REBUILDING MULTIPLE ERASURES

In this section, we discuss the rebuilding of e erasures, where 1≦e≦r. We will first prove the lower bound for rebuilding ratio and repair bandwidth. Then we show a construction achieving the lower bound for systematic nodes. At last we generalize this construction and Construction 1, and propose a rebuilding algorithm using an arbitrary subgroup and its cosets.

In this section, in order to simplify some of the results we will assume that r is a prime and the calculations are done over F_(r). Note that all the result can be generalized with minor changes for an arbitrary integer r and the ring Z_(r).

A. Lower Bounds

The next theorem shows that the rebuilding ratio for Construction 1 is at least e/r.

Theorem 25.

Let A be an array with r parity nodes constructed by Construction 5. In an erasure of 1≦e≦r systematic nodes, the rebuilding ratio is at least e/r.

Proof: In order to recover the information in the systematic nodes we need to use at least er^(m) zigzag sets from the r^(m+1) sets (There are r parity nodes, r^(m) zigzag sets in each parity). By the pigeonhole principle there is at least one parity node, such that at least er^(m−1) of its zigzag sets are used. Hence each remaining systematic node has to access its elements that are contained in these zigzag sets. Therefore each systematic node accesses at least er^(m−1) of its information out of r^(m), which is a portion of e/r.

Since we use at least er^(m) zigzag sets, we use at least er^(m) elements in the r parity nodes, which is again a portion of e/r. Hence the overall rebuilding ratio is at least e/r.

In a general code (not necessary MDS, systematic, or optimal update), what is the amount of information needed to transmit in order to rebuild e nodes? Assume that in the system multiple nodes are erased, and we rebuild these nodes simultaneously from information in the remaining nodes. It should be noted that this model is a bit different from the distributed repair problem, where the recovery of each node is done separately. We follow the definitions and notations of [N. B. Shah et al., Tech. Rep. arXiv:1011.2361v2 (2010)]. An exact-repair reconstructing code satisfies the following two properties: (i) Reconstruction: any k nodes can rebuild the total information; (ii) Exact repair: if e nodes are erased, they can be recovered exactly by transmitting information from the remaining nodes.

Suppose the total amount of information is M, and the n nodes are [n]. For e erasures, 1≦e≦r, denote by α, d_(e), β_(e) the amount of information stored in each node, the number of nodes connected to the erased nodes, and the amount of information transmitted by each of the nodes, respectively. For subsets A, B

[n], W_(A) is the amount of information stored in nodes A, and S_(A) ^(B) is the amount of information transmitted from nodes A to nodes B in the rebuilding.

The following results give lower bound of repair bandwidth for e erasures, and the proofs are based on [N. B. Shah et al., Tech. Rep. arXiv:1011.2361v2 (2010)].

Lemma 26.

Let B

[n] be a subset of nodes of size |e|, then for an arbitrary set of nodes A, |A|≦d_(e) such that B∩A=Ø,

H(W _(B) |W _(A))≦min{|B|α,(d _(e) −|A|)β_(e)}.

Proof: If nodes B are erased, consider the case of connecting to them nodes A and nodes C, |C|=d_(e)−|A|. Then the exact repair condition requires

$\begin{matrix} {0 = {H\left( {W_{B}\left. {S_{A}^{B},S_{C}^{B}} \right)} \right.}} \\ {= {H\left( {{W_{B}\left. S_{A}^{B} \right)} - {I\left( {W_{B},{{S_{C}^{B}\left. S_{A}^{B} \right)} \geq}} \right.}} \right.}} \\ {{H\left( {{{W_{B}\left. S_{A}^{B} \right)} - {H\left( S_{C}^{B} \right)}} \geq} \right.}} \\ {{H\left( {{{W_{B}\left. S_{A}^{B} \right)} - {\left( {d - {A}} \right)\beta_{e}}} \geq} \right.}} \\ {{H\left( {{W_{B}\left. W_{A} \right)} - {\left( {d - {A}} \right)\beta_{e}}} \right.}} \end{matrix}$

Moreover, it is clear that H(W_(B)|W_(A))≦H(W_(B))≦|B|α and the result follows.

Theorem 27.

Any reconstructing code with file size M must satisfy for any 1≦e≦r

$M \leq {{s\; \alpha} + {\sum\limits_{i = 0}^{{\lfloor\frac{k}{e}\rfloor} - 1}{\min \left\{ {{e\; \alpha},{\left( {d_{e} - {i\; e} - s} \right)\beta_{e}}} \right\}}}}$

where s=k mod e, 0≦s<e. Moreover for an MDS code,

$\beta_{e} \geq {\frac{e\; M}{k\left( {d - k + e} \right)}.}$

Proof: The file can be reconstructed from any set of k nodes, hence

$\begin{matrix} {M = {H\left( W_{\lbrack k\rbrack} \right)}} \\ {= {{H\left( W_{\lbrack s\rbrack} \right)} + {\sum\limits_{i = 0}^{{\lfloor\frac{k}{e}\rfloor} - 1}{H\left( {{W_{\lbrack{{{i\; e} + s + 1},{{{({i + 1})}e} + s}}\rbrack}\left. W_{\lbrack{{i\; e} + s}\rbrack} \right)} \leq} \right.}}}} \\ {{{s\; \alpha} + {\sum\limits_{i = 0}^{{\lfloor\frac{k}{e}\rfloor} - 1}{\min \left\{ {{e\; \alpha},{\left( {d_{e} - {i\; e} - s} \right)\beta_{e}}} \right\}}}}} \end{matrix}$

In an MDS code

${\alpha = \frac{M}{k}},$

hence in order to satisfy the inequality any summand of the form min {eα, (d_(e)−ie−s)β_(e)} must be at least

${e\frac{M}{k}},$

which occurs if and only if

${\left( {d_{e} - {\left( {\left\lfloor \frac{k}{e} \right\rfloor - 1} \right)e} - s} \right)\beta_{e}} \geq {\frac{eM}{k}.}$

Hence we get

$\beta_{e} \geq {\frac{eM}{k\left( {d - k + e} \right)}.}$

And the proof is completed.

Therefore, the lower bound of the repair bandwidth for an MDS code is

$\frac{eM}{k\left( {d - k + e} \right)},$

which is the same as the lower bound of the rebuilding ratio in Theorem 25.

B. Rebuilding Algorithms

Next we discuss how to rebuild in case of e erasures, 1≦e≦r, for an MDS array code with optimal update. Theorem 27 gives the lower bound e/r on the rebuilding ratio for e erasures. Is this achievable? Let us first look at an example.

Example 3

Consider the code in FIG. 5 with r=3. When e=2 and columns C₀, C₁ are erased, we can access rows {0,1,3,4,6,7} in column C₂, C₃ rows {1,2,4,5,7,8} in column C₄, and rows {2,0,5,3,8,6} in column C₅. One can check that the accessed elements are sufficient to rebuild the two erased columns, and the ratio is 2/3=e/r. It can be shown that similar rebuilding can be done for any two systematic node erasures. Therefore, in this example the lower bound is achieved.

For an array of size p×k an (n,k) MDS code with r=n−k parity nodes is constructed. Each parity node lε[0, r−1] is constructed from the set of permutations {f_(i) ^(l)} for iε[0,k−1]. Notice that in the general case the number of rows p in the array is not necessarily a power of r. We will assume columns [0, e−1] are erased. In an erasure of e columns, ep elements need rebuilt, hence we need ep equations (zigzags) that contain these elements. In an optimal rebuilding, each parity node contributes ep/r equations by accessing the values of ep/r of its zigzag elements. Moreover, the union of the zigzag sets that create these zigzag elements, constitute an e/r portion of the elements in the surviving systematic nodes. In other words, there is a set X

[0, p−1], such that |X|=ep/r and

f _(j) ^(l)(X)=f _(i) ^(l)(X)

for any parity node lε[0, r−1] and surviving systematic nodes i, jε[e, k−1]. Note that it is equivalent that for any parity node lε[0, r−1] and surviving systematic node jε[e, k−1]

f _(j) ^(l)(X)=f _(e) ^(l)(X).

Let G^(l) be the subgroup of S_(p) that is generated by the set of permutations {f_(e) ⁻¹∘f_(j) ^(l)}_(j=e) ^(k−1). It is easy to see that the previous condition is also equivalent to that for any parity lε[0, r−1] the group G¹ stabilizes X, i.e., for any fεG^(l), f(X)=X.

Assuming there is a set X that satisfies this condition, we have to guarantee that indeed the ep elements can be rebuilt from the chosen ep equations, i.e., make sure that the ep equations with the ep variables are solvable. A necessary condition is that each element in the erased column will appear at least once in the chosen zigzag sets (equations). parity lε[0, r−1] accesses its zigzag elements f_(e) ^(l)(X), and these zigzag sets contain the elements in rows (f_(i) ^(l))⁻¹f_(e) ^(l)(X) of the erased column iε[0, e−1]. Hence the condition is equivalent to that for any erased column iε[0, e−1]

∪_(l=0) ^(r−1)(f _(i) ^(l))⁻¹ f _(e) ^(l)(X)=[0,p−1].

These two conditions are necessary for optimal rebuilding ratio. In addition, we need to make sure that the ep equations are linearly independent, which it depends on the coefficients in the linear combinations that created the zigzag elements. These three conditions are necessary and sufficient conditions for optimal rebuilding ratio of e erasures. We summarize:

Sufficient and necessary conditions for optimal rebuilding ratio in e erasures:

There exists a set X

[0, p−1] of size |X|=ep/r, such that

(1) For any parity node lε[0, e−1] the group G^(l) stabilizes the set X, i.e., for any gεG^(l)

g(X)=X.  (34)

where G^(l) is generated by the set of permutations {f_(e) ⁻¹ ∘f _(j) ^(l)}_(j=e) ^(k−1).

(2) For any erased column iε[0, e−1],

∪_(l=0) ^(r−1)(f _(i) ^(l))⁻¹ f _(e) ^(l)(X)=[0,p−1].  (35)

(3) The ep equations (zigzag sets) defined by the set X are linearly independent.

The previous discussion gave the condition for optimal rebuilding ratio in an MDS optimal update code with e erasures. Next will interpret these conditions in the special case where the number of rows p=r^(m), and the permutations are generated by T={v₀, v₁, . . . , v_(k−1)}

F_(r) ^(m) and Construction 5, i.e., f_(i) ^(l)(x)=x+lv_(i) for any xε[0, r^(m)−1]. Note that in the case of r a prime

G ¹ =G ² = . . . =G ^(r−1),

and in that case we simply denote the group as G. The following theorem gives a simple characterization for sets that satisfy condition 1.

Theorem 28.

Let X

F_(r) ^(m) and G defined above then G stabilizes X, if and only if X is a union of cosets of the subspace

Z=span{v _(e+1) −v _(e) , . . . , v _(k−1) −v _(e)}.  (36)

Proof: It is easy to check that any coset of Z is stabilized by G, hence if X is a union of cosets it is also a stabilized set. For the other direction let x, yεF_(r) ^(m) be two vectors in the same coset of Z, it is enough to show that if xεX then also yeX. Since y−xεZ there exist a₁, . . . , a_(k−1−e)ε[0, r−1] such that y−x=Σ_(i−1) ^(k−1−e)α_(i)(v_(e+i)−v_(e)). Since f(X)=X for any fεG we get that f(x)εX for any xεX and fεG, hence

$\begin{matrix} {y = {x + y - x}} \\ {= {x + {\sum\limits_{i = 1}^{k - 1 - e}{\alpha_{i}\left( {v_{e + i} - v_{e}} \right)}}}} \\ {{= {f_{e}^{- \alpha_{k - 1 - e}}f_{k - 1}^{\alpha_{k - 1 - e}}\mspace{14mu} \ldots \mspace{14mu} f_{e}^{- \alpha_{1}}{f_{e + 1}^{\alpha_{1}}(x)}}},} \end{matrix}$

for f_(e) ^(−α) ^(k−1−e) f_(k−1) ^(α) ^(k−1−e) . . . f_(e) ^(−α) ¹ f_(e+1) ^(α) ¹ εG and the result follows.

Remark: For any set of vectors S and v, uεS,

span{S−v}=span{S−u}.

Where, S−v={v_(i)−v|v_(i)εS}. Hence, the subspace Z defined in the previous theorem does not depend on the choice of the vector v_(e). By the previous theorem we interpret the necessary and sufficient conditions of an optimal code as follows:

There exists a set X

F_(r) ^(m) of size |X|=er^(m−1), such that

(1) X is a union of cosets of

Z=span{v _(e+1) −v _(e) , . . . , v _(k−1) −v _(e)}.

(2) For any erased column iε[0, e−1],

∪_(l=0) ^(r−1)(X+l(v _(i) −v _(e)))=F _(r) ^(m).  (37)

(3) The er^(m) equations (zigzag sets) defined by the set X are linearly independent.

The following theorem gives a simple equivalent condition for conditions 1, 2.

Theorem 29.

There exists a set X

F_(r) ^(m) of size |X|=er^(m−1 such that conditions) 1, 2 are satisfied if and only if

v _(i) −v _(e) εZ,  (38)

for any erased column iε[0, e−1].

Proof: Assume conditions 1, 2 are satisfied. If v_(i)−v_(e)εZ for some erased column iε[0, e−1] then X∪_(l=0) ^(r−1)(X+l(v_(i)−v_(e)))=F_(r) ^(m). Which is a contradiction to that X⊂F_(r) ^(m). On the other hand, if Equation (38) is true, then v_(i)−v_(e) can be viewed as a permutation that acts on the cosets of Z. The number of cosets of Z is r^(m)/|Z| and this permutation (when it is written in cycle notation) contains r^(m−1)/|Z| cycles, each with length r. For each iε[0, e−1] choose r^(m−1)/|Z| cosets of Z, one from each cycle of the permutation v_(i)−v_(e). In total er^(m−1)/|Z| cosets are chosen for the e erased nodes. Let X be the union of the cosets that were chosen. It is easy to see that X satisfies condition 2 above. If |X|<er^(m−1) (Since there might be cosets that were chosen more than once) add arbitrary (er^(m−1)−|X|)/|Z| other cosets of Z, and also condition 1 is satisfied.

In general, if Equation (38) is not satisfied, the code does not have an optimal rebuilding ratio. However we can define

Z=span{v _(i) −v _(e)}_(iεI),  (39)

where we assume w.l.o.g. eεI and I

[e, k−1] is a maximal subset of surviving nodes that satisfies for any erased node jε[0, e−1], v_(j)−v_(e)εZ. Hence from now on we assume that Z is defined by a subset of surviving nodes I. This set of surviving nodes will have an optimal rebuilding ratio (see Corollary 32), i.e., in the rebuilding of columns [0, e−1], columns I will access a portion of e/r of their elements. The following theorem gives a sufficient condition for the er^(m) equations defined by the set X to be solvable linear equations.

Theorem 30.

Suppose that there exists a subspace X₀ that contains Z such that for any erased node iε[0, e−1]

X ₀⊕span{v _(i) −v _(e) }=F _(r) ^(m),  (40)

then the set X defined as an union of some e cosets of X₀ satisfies conditions 1, 2 and 3 over a field large enough.

Proof: Condition 1 is trivial. Note that by Equation (40) above, l(v_(i)−v_(e))∉X₀ for any lε[1, r−1] and iε[0, e−1], hence {X₀+l(v_(i)−v_(e))}_(lε[0,r−1]) is the set of cosets of X₀. Let X_(j)=X₀+j(v_(i)−v_(e)) be a coset of X₀ for some iε[0, e−1] and suppose X_(j)⊂X.

$\begin{matrix} {{{\bigcup_{l = 0}^{r - 1}\left( {X + {l\left( {v_{i} - v_{e}} \right)}} \right)} \supseteq {\bigcup_{l = 0}^{r - 1}\left( {X_{j} + {l\left( {v_{i} - v_{e}} \right)}} \right)}} = {{\bigcup_{l = 0}^{r - 1}\left( {X_{0} + {j\left( {v_{i} - v_{e}} \right)} + {l\left( {v_{i} - v_{e}} \right)}} \right)} = {{\bigcup_{l = 0}^{r - 1}\left( {X_{0} + {\left( {j + l} \right)\left( {v_{i} - v_{e}} \right)}} \right)} = {\bigcup_{t = 0}^{r - 1}\left( {X_{0} + {t\left( {v_{i} - v_{e}} \right)}} \right)}}}} & (41) \\ {\mspace{20mu} {= {_{r}^{m}.}}} & (42) \end{matrix}$

Equation (41) holds, since j+l is computed mod r. So condition 2 is satisfied. Next we prove condition 3. There are er^(m) unknowns and er^(m) equations. Writing the equations in a matrix form we get AY=b, where A is an er^(m)×er^(m) matrix. Y, b are vectors of length er^(m), and Y=(y₁, . . . , y_(er) _(m) )^(T) is the unknowns vector. The matrix A=(a_(i,j)) is defined as a_(i,j)=x_(i,j) if the unknown y_(j) appears in the i-th equation, otherwise a_(i,j)=0. Hence we can solve the equations if and only if there is assignment for the indetermediates {x_(i,j)} in the matrix A such that det(A)≠0. By Equation (42), accessing rows corresponding to any coset X_(j) will give us equations where each unknown appears exactly once. Since Xis a union of e cosets, each unkown appears e times in the equations. Thus each column in A contains e indeterminates. Moreover, each equation contains one unknown from each erased node, thus any row in A contains e indeterminates. Then by Hall's marriage Theorem [P. Hall, J. London Math. Soc., 10(1):26-30 (1935)] we conclude that there exists a permutation f on the integers [1, er^(m)] such that

${\prod\limits_{i = 1}^{{er}^{m}}a_{i,{f{(i)}}}} \neq 0.$

Hence the polynomial det(A) when viewed as a symbolic polynomial, is not the zero polynomial, i.e.,

${\det (A)} = {{\sum\limits_{f \in S_{{er}^{m}}}{{{sgn}(f)}{\prod\limits_{i = 1}^{{er}^{m}}a_{i,{f{(i)}}}}}} \neq 0.}$

By Theorem 2 we conclude that there is an assignment from a field large enough for the indeterminates such that det(A)≠0, and the equations are solvable. Note that this proof is for a specific set of erased nodes. However if Equation (40) is satisfied for any set of e erasures, multiplication of all the nonzero polynomials det(A) derived for any set of erased nodes is again a nonzero polynomial and by the same argument there is an assignment over a field large enough such that any of the matrices A is invertible, and the result follows.

In order to use Theorem 30, we need to find a subspace X₀ as in Equation (40). The following Theorem shows that such a subspace always exists, moreover it gives an explicit construction of it.

Theorem 31.

Suppose 1≦e<r erasures occur. Let Z be defined by Equation (39) and v_(i)−v_(e)εZ for any erased node iε[0, e−1]. Then there exists u⊥Z such that for any iε[0, e−1],

u·(v _(i) −v _(e))≠0.  (43)

Moreover the orthogonal subspace X₀=(u)^(⊥) satisfies Equation (40).

Proof: First we will show that such vector u exists. Let u₁, . . . u_(t) be a basis for (Z)^(⊥) the orthogonal subspace of Z. Any vector u in (Z)^(⊥) can be written as u=Σ_(j=1) ^(t)x_(j)u_(j) for some x_(j)'s. We claim that for any iε[0, e−1] there exists j such that u_(j)·(v_(i)−v_(e))≠0. Because otherwise, (Z)^(⊥)=span{u₁, . . . u_(t)}⊥v_(i)−v_(e), which means v_(i)−v_(e)εZ and reaches a contradiction. Thus the number of solutions for the linear equation

${\sum\limits_{j = 1}^{t}{x_{j}{u_{j} \cdot \left( {v_{i} - v_{e}} \right)}}} = 0$

is r^(t−1), which equals the number of u such that u·(v_(i)−v_(e))=0. Hence by the union bound there are at most er^(t−1) vectors u in (Z)^(⊥) such that u·(v_(i)−v_(e))=0 for some erased node iε[0, e−1]. Since |(Z)^(⊥)|=r^(t)>er^(t−1) there exists u in (Z)^(⊥) such that for any erased node iε[0, e−1],

u·(v _(i) −v _(e))≠0.

Define X₀=(u)^(⊥), and note that for any erased node iε[0, e−1], v_(i)−v_(e)∉X₀, since u·(v_(i)−v_(e))≠0 and X₀ is the orthogonal subspace of u. Moreover, since X₀ is a hyperplane we conclude that

X ₀⊕span{v _(i) −v _(e) }=F _(r) ^(m),

and the result follows.

Theorems 30 and 31 give us an algorithm to rebuild multiple erasures:

-   -   (1) Find Z by Equation (34) satisfying Equation (33).     -   2) Find u⊥Z satisfying Equation (38); define X₀=(u)⊥ and X as a         union of e cosets of X₀.     -   3) Access rows f_(e) ^(l)(X) in parity iε[0, r−1] and all the         corresponding information elements.         We know that under a proper selection of coefficients the         rebuilding is possible.

In the following we give two examples of rebuilding using this algorithm. The first example shows an optimal rebuilding for any set of e node erasures. As mentioned above, the optimal rebuilding is achieved since Equation (38) is satisfied, i.e., I=[e, k−1].

Example 4

Let T={v₀, v₁, . . . , v_(m)} be a set of vectors that contains an orthonormal basis of F_(r) ^(m) together with the zero vector. Suppose columns [0, e−1] are erased. Note that in that case I=[e, k−1] and Z is defined as in Equation (39). Define

${u = {\sum\limits_{j = e}^{m}v_{j}}},$

and X₀=(u)^(⊥). When m=r and e=r−1, modify u to be

$u = {\sum\limits_{i = 1}^{m}{v_{i}.}}$

It is easy to check that u⊥Z and for any erased column iε[0, e−1], u·(v_(i)−v_(e))=−1. Therefore by Theorems 30 and 31 a set X defined as a union of an arbitrary e cosets of X₀ satisfies conditions 1, 2, and 3, and optimal rebuilding is achieved.

In the example of FIG. 7, we know that the vectors generating the permutations are the standard basis (and thus are orthonormal basis) and the zero vector. When columns C₀, C₁ are erased, u=e₂ and X₀=(u)^(⊥)=span{e₁}={0,3,6}. Take X as the union of X₀ and its coset {1,4,7}, which is the same as Example 3. One can check that each erased element appears exactly three times in the equations and the equations are solvable in F₇. Similarly, the equations are solvable for other two systematic erasures.

Before we proceed to the next example, we give an upper bound for the rebuilding ratio using Theorem 30 and a set of nodes I.

Corollary 32. Theorem 30 requires rebuilding ratio at most

$\frac{e}{r} + \frac{\left( {r - e} \right)\left( {k - {I} - e} \right)}{r\left( {k + r - e} \right)}$

Proof: By Theorem 29, the fraction of accessed elements in columns I and the parity columns is e/r of each column. Moreover, the accessed elements in the rest columns are at most an entire column. Therefore, the ratio is at most

$\frac{{\frac{e}{r}\left( {{I} + r} \right)} + \left( {k - {I} - e} \right)}{k + r - e} = {\frac{e}{r} + \frac{\left( {r - e} \right)\left( {k - {I} - e} \right)}{r\left( {k + r - e} \right)}}$

and the result follows.

Note that as expected when |I|=k−e the rebuilding ratio is optimal, i.e. e/r. In the following example the code has O(m²) columns. The set I does not contain all the surviving systematic nodes, hence the rebuilding is not optimal but is at most

$\frac{1}{2} + {{O\left( \frac{1}{m} \right)}.}$

Example 5

Suppose 2|m. Let T={v=(v₁, . . . , v_(m)):∥v∥₁=2, v_(i)=1, v_(j)=1, for some iε[1, m/2], jε[m/2+1, m]}⊂F_(r) ^(m) be the set of vectors generating the code with r=2 parities, hence the number of systematic nodes is |T|=k=m²/4. Suppose column w={w₁, . . . , w_(m)}, w₁=w_(m/2+1)=1 was erased. Define the set I={vεT:v₁=0}, and

Z=span{v ₁ −v _(e) |iεI}

for some eεI. Thus |I|=m(m−2)/4. It can be seen that Z defined by the set I satisfies Equation (38), i.e., w−v_(e)εZ since the first coordinate of a vector in Z is always 0, as opposed to 1 for the vector w−v_(e). Define u=(0, 1, . . . , 1) and X₀=(u)^(⊥). It is easy to check that u⊥Z and u·(w−v_(e))=1≠0. Hence, the conditions in Theorem (31) are satisfied and a rebuilding can be done using one of the two cosets of x₀. Moreover, by Corollary 32 the rebuilding ratio is at most

${{\frac{1}{2} + {\frac{1}{2}\frac{\left( {m/2} \right) - 1}{\left( {m^{2}/4} \right) + 1}}} \approx {\frac{1}{2} + \frac{1}{m}}},$

which is a little better than Theorem 18 in the constants. Note that by similar coefficients assignment of Construction 4, we can use a field of size five or eight to assure the code will be an MDS code.

C. Minimum Number of Erasures with Optimal Rebuilding

Next we want to point out a surprising phenomenon. We say that a set of vectors S satisfies property e for e≧1 if for any subset A

S of size e and any eεA,

u−vεspan{w−v:wεS\A},

where vεS\A. Recall that by Theorem 29 any set of vectors that generates a code C and can rebuild optimally any e erasures, satisfies property e. The following theorem shows that this property is monotonic, i.e., if S satisfies property e then it also satisfies property a for any e≦a≦|S|.

Theorem 33.

Let S be a set of vectors that satisfies property e, then it also satisfies property a, for any e≦a≦|s|.

Proof: Let A

S, |A|=e+1 and assume to the contrary that u−vεspan{w−v:wεS\A} for some uεA and vεS\A. |A|≧2 hence there exists xεA\u. It is easy to verify that u−vεspan{w−v:wεS\A*}, where A*=A\x and |A*|=e which contradicts the property e for the set S.

Hence, from the previous theorem we conclude that a code C that can rebuild optimally e erasures, is able to rebuild optimally any number of erasures greater than e as well. However, as pointed out already there are codes with r parities that can not rebuild optimally from some e<r erasures. Therefore, one might expect to find a code C with parameter e*≧1 such that it can rebuild optimally e erasures only when e*≦e≦r. For example, for r=3, m=2 let C be the code constructed by the vectors {0, e₁, e₂, e₁+e₂}. We know that any code with more than 3 systematic nodes cannot rebuild one erasure optimally, since the size of a family of orthogonal permutations over the integers [0,3²−1] is at most 3. However, one can check that for any two erased columns, the conditions in Theorem 30 are satisfied hence the code can rebuild optimally for any e=2 erasures and we conclude that e*=2 for this code.

The phenomena that some codes has a threshold parameter e*, such that only when the number of erasures e is at least as the threshold e* then the code can rebuild optimally, is a bit counter intuitive and surprising. This phenomena gives rise to another question. We know that for a code constructed with vectors from F_(r) ^(m), the maximum number of systematic columns for optimal rebuilding of e=1 erasures is m+1 (Theorem 23). Can the number of systematic columns in a code with an optimal rebuilding of e>1 erasures be increased? The previous example shows a code with four systematic columns can rebuild optimally any e=2 erasures. But Theorem 23 shows that when r=3, m=2, optimal rebuilding for one erasure implies no more than three systematic columns. Hence the number of systematic columns is increased by at least 1 compared to codes with nine rows and optimal rebuilding of one erasure. The following theorem gives an upper bound for the maximum systematic columns in a code that rebuilds optimally any e erasures.

Theorem 34.

Let C be a code constructed by Construction 5 and vectors from F_(r) ^(m). If C can rebuild optimally any e erasures, for some 1≦e<r, then the number of systematic columns k in the code satisfies

k≦m+e.

Proof: Consider a code with length k and generated by vectors v₀, v₁, . . . , v_(k−1). If these vectors are linearly independent then k≦m and we are done. Otherwise they are dependent. Suppose e columns are erased, 1≦e<r. Let v_(e) ye be a surviving column. Consider a new set a of vectors: T={v_(i)−v_(e):iε[0,k−1], i≠e}. We know that the code can rebuild optimally only if Equation (38) is satisfied for all possible e erasures. Thus for any i≠e, iε[0,k−1], if column i is erased and column e is not, we have v_(i)−v_(e)εZ and thus v_(i)−v_(e)≠0. So every vector in T is nonzero. Let s be the minimum number of dependent vectors in T, that is, the minimum number of vectors in T such that they are dependent. For nonzero vectors, we have s≧2. Say {v_(e+1)−v_(e), v_(e+2)−v_(e), . . . , v_(e+s)−v_(e)} is a minimum dependent set of vector. Since any m+1 vectors are dependent in F_(r) ^(m),

s≦m+1.

We are going to show k−e≦s−1. Suppose to the contrary that the number of remaining columns satisfies k−e≧s and e erasures occur. When column v_(e+s) is erased and the s columns {v_(e), v_(e+1), . . . , v_(e+s−1)} are not, we should be able to rebuild optimally. However since we chose a dependent set of vectors, v_(e+s)−v_(e) is a linear combination of {v_(e+1)−v_(e), v_(e+2)−v_(e), . . . , v_(e+s−1)−v_(e)}, whose span is contained in Z in Equation (38). Hence Equation (38) is violated and we reach a contradiction. Therefore,

k−e≦s−1≦m.

Notice that this upper bound is tight. For e=1 we already gave codes with optimal rebuilding of one erasure and k=m+1 systematic columns. Moreover, for e=2 the code already presented in this section and is constructed by the vectors 0, e₁, e₂, e₁+e₂, reaches the upper bound with k=4 systematic columns.

D. Generalized Rebuilding Algorithms

The rebuilding algorithms presented in Constructions 1 and 5 and in Theorem 30 all use a specific subspace and its cosets in the rebuilding process. This method of rebuilding can be generalized by using an arbitrary subspace as explained below.

Let T={v₀, . . . , v_(k−1)} be a set of vectors generating the code in Construction 5 with r^(m) rows and r parities. Suppose e columns [0, e−1] are erased. Let Z be a proper subspace of F_(r) ^(m). In order to rebuild the erased nodes, in each parity column lε[0, r−1], access the zigzag elements z_(i) ^(l) for iεX_(l), and X_(l) is a union of cosets of Z. In each surviving node, access all the elements that are in the zigzag sets X_(l) of parity l. More specifically, access element a_(i,j) in the surviving column jε[e, k−1] if i+lv_(j)εX_(l). Hence, in the surviving column j and parity l, we access elements in rows X₁−lv_(j). In order to make the rebuilding successful we impose the following conditions on the sets X₀, . . . , X_(l). Since the number of equations needed is at least as the number of erased elements, we require

$\begin{matrix} {{\sum\limits_{l = 0}^{r - 1}{X_{l}}} = {{er}^{m}.}} & (44) \end{matrix}$

Moreover, we want the equations to be solvable, hence for any erased column iε[0, e−1],

∪_(l=0) ^(r−1) X _(l) −lv _(i)=[0,r ^(m)−1] multiplicity e,  (45)

which means if the union is viewed as a multi-set, then each element in [0, r^(m)−1] appears exactly e times. This condition makes sure that the equations are solvable by Hall's theorem (see Theorem 30). Under these conditions we would like to minimize the ratio, i.e., the number of accesses which is,

$\begin{matrix} {\min\limits_{X_{0},\ldots \mspace{11mu},X_{r - 1}}{\sum\limits_{j = e}^{k - 1}{{{\bigcup_{l = 0}^{r - 1}\left( {X_{l} = {lv}_{j}} \right)}}.}}} & (46) \end{matrix}$

In summary, for the generalized rebuilding algorithm one first chooses a subspace Z, and then solves the minimization problem in (46) subject to (44) and (45).

The following example interprets the minimization problem for a specific case.

Example 6

Let r=2, e=1, i.e., two parities and one erasure, then Equations (44), (45) become

|X ₀ |+|X ₁|=2^(m) ,X ₀ ∪X ₁ +v ₀=[0,2^(m)−1].

Therefore X₁+v₀= X₀ . The objective function in Equation (46) becomes,

${\min\limits_{X_{0},X_{1\;}}{\sum\limits_{j = 1}^{k - 1}{{X_{0}\bigcup{X_{1} + v_{j}}}}}} = {\min\limits_{X_{0}}{\sum\limits_{j = 1}^{k - 1}{{{X_{0}\bigcup\left( {\overset{\_}{X_{0}} + v_{0} + v_{j}} \right)}}.}}}$

Each v₀+v_(j) defines a permutation f_(v) ₀ _(+v) _(j) on the cosets of Z by f_(v) ₀ _(+v) _(j) (A)=A+v₀+v_(j) for a coset A of Z. If v₀+v_(j)εZ then f_(v) ₀ _(+v) _(j) is the identity permutation and |X₀∪( X₀ +v₀+v_(j))|=2^(m), regardless of the choice of X₀. However, if v₀+v_(j)∉Z , then f_(v) ₀ _(+v) _(j) is of order 2, i.e., it's composed of disjoint cycles of length 2. Note that if f_(v) ₀ _(+v) _(j) maps A to B and only one of the cosets A, B is contained in X₀, say A, then only A is contained in X₀∪( X₀ +v₀+v_(j)). On the other hand, if both A, BεX₀ or A, B∉X₀ then,

A,B

X ₀∪( X ₀ +v ₀ +v _(j)).

In other words, (A, B) is a cycle in f_(v) ₀ _(+v) _(j) which is totally contained in X₀ or in X₀ . Define N_(j) ^(X) as the number of cycles (A, B) in the permutation f_(v) ₀ _(+v) _(j) that are totally contained in X or in X, where X is a union of some cosets of Z. It is easy to see that the minimization problem is equivalent to minimizing

$\begin{matrix} {\min\limits_{X}{\sum\limits_{j = 1}^{k - 1}\; {N_{j}^{X}.}}} & (47) \end{matrix}$

In other words, we want to find a set X which is a union of cosets of Z, such that the number of totally contained or totally not contained cycles in the permutations defined by v_(j)+v₀, jε[1, k−1] is minimized.

From the above example, we can see that given a non-optimal code with two parities and one erasure, finding the solution in Equation (47) requires minimizing for the sum of these k−1 permutations, which is an interesting combinatorial problem. Moreover, by choosing a different subspace Z we might be able to get a better rebuilding algorithm than that in Construction 1 or Theorem 30.

IX. PHYSICAL IMPLEMENTATIONS

In this discussion, we described explicit constructions of the first known systematic (n,k) MDS array codes with n−k equal to some constant, such that the amount of information needed to rebuild an erased column equals to 1/(n−k), which matches the information-theoretic lower bound. Codes constructed according to these principles may be utilized in RAID storage arrays through implementation in RAID controllers. Such RAID controllers will generate parity data according to the code principles described herein and will perform read, write, and rebuilding operations using such codes. The structure and operation of such controllers will be described next, for purposes of illustration.

A. RAID Controller

FIG. 8 shows a block diagram of a RAID storage system 800 constructed in accordance with the disclosure. In the system 800, a host computer 802 communicates with a RAID controller 804 to write and read data to and from a RAID array 806. The RAID array comprises multiple systematic nodes 808 a, 808 b, 808 c of user data and multiple parity nodes 810 a, 810 b of parity data. The nodes will be referred to collectively by their numerals, without letter suffix, unless identification of particular nodes is desired. Each of the systematic nodes 808 and parity nodes 810 may comprise a single disk drive, or multiple drives, or RAID arrays. The collection of systematic nodes 808 and parity nodes 810 may comprise one or more nodes, in any number according to the system resources, as indicated by the ellipses in FIG. 8.

The RAID controller 804 communicates with the host computer 802 over a host transport medium 812 through a host interface 814. The host transport medium may comprise, for example, a network connection that is wired or wireless, or it may comprise a system connection such as a system bus or the like. The host interface may comprise, for example, a network interface card or other means for communications. The RAID controller 804 communicates with the nodes 808, 810 of the RAID array 806 over an array transport medium 818 through a node interface 816. The array transport medium may comprise, for example, a network connection that is wired or wireless, or may comprise a system connection such as a system bus or the like. The node interface may comprise, for example, a network interface card or other means for communications.

The RAID controller 804 comprises a computer device and, as described further below, as such the RAID controller includes memory and a central processor unit, providing an operating system of the RAID controller. A controller application 820 operates in the RAID controller, supported by the operating system of the RAID controller, such that the controller application manages communications through the host interface 814 and the node interface 816, and also manages code generation for the write and read operations with the RAID array and any rebuilding operations upon node erasure.

FIG. 9 shows a flow diagram of the operations performed by the RAID controller 804 of FIG. 8. In the operation indicated by box 902 of FIG. 9, the RAID controller receives RAID configuration data for the RAID array. The configuration data includes, for example, a specification of the nodes in the array, including the location of the nodes, the storage capacity of the nodes, the number of nodes, and other information necessary for the RAID controller to configure and manage operations for the RAID array.

After the RAID configuration data has been received, the RAID controller can determine the parity code to be used for the RAID array operations at box 904. The determination of the code is performed in accordance with the principles in this disclosure. The particular code to be used will be selected in accordance with system resources and in accordance with any predetermined features that are implemented in the controller application.

After the configuration data has been received and the RAID controller has determined the parity code to be used, the read and write data operations with the RAID array may be performed by the RAID controller. Such operations are represented by the next box 906 in FIG. 9. If an erasure occurs, such as a node failure, then the RAID controller detects the event and takes corrective action according to the decision box 908. The RAID controller may indicate a node erasure if, for example, a write failure or read failure occurs. Upon detecting the erasure, an affirmative outcome at the box 908, then the RAID controller performs a rebuild operation as described above, in accordance with the principles of this disclosure. Performing the rebuild operation is indicated at the box 910. If no erasure event is detected, a negative outcome at the decision box 908, then the RAID controller continues with read and write operations of the RAID array, as indicated by the return of controller operations from the decision box 908 to reading and writing at the box 906.

It should be understood that the operation of receiving RAID configuration data (box 902 of FIG. 9) may be performed upon initiation by a user through the host computer. That is, configuration of the RAID array need not await an initial setup operation or a node failure. Rather, configuration of the RAID array according to the controller application may be initiated upon user command, with operations to follow as illustrated in FIG. 9.

B. Computer System

FIG. 10 shows a block diagram of a computer system such as utilized as either or both of the RAID controller and host computer of FIG. 8.

The operations described above for operating a RAID array, for reading data from a RAID array, for writing data to a RAID array, and for rebuilding a RAID array, can be carried out by the operations depicted in FIG. 9, which can be performed by the controller application 820 and associated components of the RAID controller 804 illustrated in FIG. 8. The controller application may be installed in a variety of computer devices that control and manage operations of an associated RAID array. For example, in an implementation of the coding scheme described herein within a single controller device, all the components of the RAID controller 804 depicted in FIG. 8 can be contained within firmware of a controller device that communicates with a host computer and RAID nodes.

The processing components such as the controller application 820 and host interface 814 and node interface 816 may be implemented in the form of control logic in software or hardware or a combination of both, and may comprise processors that execute software program instructions from program memory, or as firmware, or the like. The host computer 802 may comprise a conventional computer apparatus. A conventional computer apparatus also may carry out the operations of FIG. 9. For example, all the components of the RAID controller can be provided by applications that are installed on the computer system illustrated in FIG. 10. FIG. 10 is a block diagram of a computer apparatus 1000 sufficient to perform as a host computer and a RAID controller, and sufficient to perform the operations of FIG. 9.

FIG. 10 is a block diagram of a computer system 1000 that may incorporate embodiments of the present invention and perform the operations described herein. The computer system 1000 typically includes one or more processors 1005, a system bus 1010, storage subsystem 1015 that includes a memory subsystem 1020 and a file storage subsystem 1025, user interface output devices 1030, user interface input devices 1035, a communications subsystem 1040, and the like.

In various embodiments, the computer system 1000 typically includes conventional computer components such as the one or more processors 1005. The file storage subsystem 1025 can include a variety of memory storage devices, such as a read only memory (ROM) 1045 and random access memory (RAM) 1050 in the memory subsystem 1020, and direct access storage devices such as disk drives.

The user interface output devices 1030 can comprise a variety of devices including flat panel displays, touchscreens, indicator lights, audio devices, force feedback devices, and the like. The user interface input devices 1035 can comprise a variety of devices including a computer mouse, trackball, trackpad, joystick, wireless remote, drawing tablet, voice command system, eye tracking system, and the like. The user interface input devices 1035 typically allow a user to select objects, icons, text and the like that appear on the user interface output devices 1030 via a command such as a click of a button or the like.

Embodiments of the communication subsystem 1040 typically include an Ethernet card, a modem (telephone, satellite, cable, ISDN), (asynchronous) digital subscriber line (DSL) unit, FireWire (IEEE 1394) interface, USB interface, and the like. For example, the communications subsystem 1040 may be coupled to communications networks and other external systems 1055 (e.g., a network such as a LAN or the Internet), to a FireWire bus, or the like. In other embodiments, the communications subsystem 1040 may be physically integrated on the motherboard of the computer system 1000, may be a software program, such as soft DSL, or the like.

The RAM 1050 and the file storage subsystem 1025 are examples of tangible media configured to store data such as RAID configuration data, codewords, and program instructions to perform the operations described herein when executed by the one or more processors, including executable computer code, human readable code, or the like. Other types of tangible media include program product media such as floppy disks, removable hard disks, optical storage media such as CDs, DVDs, and bar code media, semiconductor memories such as flash memories, read-only-memories (ROMs), battery-backed volatile memories, networked storage devices, and the like. The file storage subsystem 1025 includes reader subsystems that can transfer data from the program product media to the storage subsystem 1015 for operation and execution by the processors 1005.

The computer system 1000 may also include software that enables communications over a network (e.g., the communications network 1055) such as the DNS, TCP/IP, UDP/IP, and HTTP/HTTPS protocols, and the like. In alternative embodiments, other communications software and transfer protocols may also be used, for example IPX, or the like.

It will be readily apparent to one of ordinary skill in the art that many other hardware and software configurations are suitable for use with the present invention. For example, the computer system 1000 may be a desktop, portable, rack-mounted, or tablet configuration. Additionally, the computer system 1000 may be a series of networked computers. Further, a variety of microprocessors are contemplated and are suitable for the one or more processors 1005, such as microprocessors from Intel Corporation of Santa Clara, Calif., USA; microprocessors from Advanced Micro Devices, Inc. of Sunnyvale, Calif., USA; and the like. Further, a variety of operating systems are contemplated and are suitable, such as WINDOWS. XP, WINDOWS 7, or the like from Microsoft Corporation of Redmond, Wash., USA, SOLARIS from Sun Microsystems, Inc. of Santa Clara, Calif., USA, various Linux and UNIX distributions, and the like. In still other embodiments, the techniques described above may be implemented upon a chip or an auxiliary processing board (e.g., a programmable logic device or graphics processor unit).

The present invention can be implemented in the form of control logic in software or hardware or a combination of both. The control logic may be stored in an information storage medium as a plurality of instructions adapted to direct an information-processing device to perform a set of steps disclosed in embodiments of the present invention. Based on the disclosure and teachings provided herein, a person of ordinary skill in the art will appreciate other ways and/or methods to implement the present invention.

X. CONCLUDING REMARKS

In this paper, we described explicit constructions of the first known systematic (n,k) MDS array codes with n−k equal to some constant, such that the amount of information needed to rebuild an erased column equals to 1/(n−k), which matches the information-theoretic lower bound.

Also, it is interesting to consider the ratio of read accesses in the case of a write (update) operation. For example, in an array code with 2 redundancies, in order to update a single information element, one needs to read at least 3 times and write 3 times, because we need to know the values of the old information and old parity and compute the new parity element (by subtracting the old information from the parity and adding the new information). However, in our optimal code construction with 2 redundancies, if we update all the information in column 1 and the rows in the first half of the array (see FIG. 3), we do not need to read anything since we know the values of all the information needed for computing the parity. These information elements take about half the size of the entire array. So in a storage system we can cache the information to be written until most of these half elements need update (we could arrange the information in a way that these elements are often updated at the same time), and the number of reading operations compared to the information size is very small. And similarly we can do the same for any other systematic column. In general, given r redundancies, we could avoid reading operations if we update about 1/r of the array.

We note that one can add redundancy for the sake of lowering the rebuilding ratio. For instance, one can use three redundancies instead of two. The third parity is not used for protecting data from erasure, because in a lot of storage systems, three failures at the same time is unlikely. But with three redundancies, we are able to access only 1/3 of data instead of 1/2 and rebuild one node failure. One open problem would be to find codes with three redundancies such that the first two redundancy ensures a rebuilding ratio of 1/2 and the third redundancy further lowers the ratio. Thus, we can build a code with two redundancies first and when cost permitted, build an extra redundancy node and obtain extra gains in the ratio, which provides flexibility in practice.

Moreover, we do not know for a given array size, what the ratio is for a code defined by arbitrary permutations. In Theorem 14, we showed that 1/2+1/k is optimal for code constructed by binary vectors and duplication. However, the ratio is not known for arbitrary permutations. Finally, it is interesting to understand how to perform efficient rebuilding in the case of two erasures for array codes that can correct three or more erasures. 

1. A computer method of operating a controller of an array of storage nodes, the method comprising: receiving configuration data at the controller that indicates operating features of the array; and determining a parity code for operation of the array according to a permutation, wherein the configuration data specifies the array as comprising nodes defined by A=(a_(i,j)) with size r^(m)×k for some integers k, m, and wherein for T={v₀, . . . , v_(k−1)}

Z_(r) ^(m) a subset of vectors of size k, where for each v=(v₁, . . . , v_(m))εT, gcd(v₁, . . . , v_(m), r), where gcd is the greatest common divisor, such that for any l, 0≦l≦r−1, and vεT, the code values are determined by the permutation f_(v) ^(l):[0, r^(m)−1]→[0, r^(m)−1] by f_(v) ^(l)(x)=x+lv.
 2. The computer method of claim 1, wherein A specified by the configuration data comprises A=(a_(i,j)), an array of size 2^(m)×k for some integers k, m, and k≦2^(m), and wherein for T

F₂ ^(m) is a subset of vectors of size k that does not contain the zero vector, and for vεT the permutation is given by f_(v):[0,2^(m)−1]→[0,2^(m)−1] by f_(v)(x)=x+v, where x is represented in its binary representation.
 3. The computer method of claim 2, wherein for the permutations f₀, . . . , f_(m) and for sets X₀, . . . , X_(m) constructed by the vectors {e_(i)}_(i=0) ^(m), the set X₀ is modified to be X₀={xεF₂ ^(m):x·(1, 1, . . . , 1)=0}.
 4. The computer method of claim 2, further comprising generating an s-duplication code of the code values, such that generating the s-duplication code comprises assigning a_(i,j) ^((t))=1 for all i, j, t, such that for odd q, and s—q−1 and assigning for all tε[0, s−1] $\beta_{i,j}^{(t)} = \left\{ \begin{matrix} {a^{t + 1},} & {{{if}\mspace{14mu} {u_{j} \cdot i}} = 1} \\ {a^{t},} & {o.w.} \end{matrix} \right.$ where u_(j)=Σ_(l=0) ^(j)e_(l) and for even q (powers of 2), and s≦q−2 and assigning for all tε[0, s−1] $\beta_{i,j}^{(t)} = \left\{ \begin{matrix} {a^{{- t} - 1},} & {{{if}\mspace{14mu} {u_{j} \cdot i}} = 1} \\ {a^{t + 1},} & {{o.w}..} \end{matrix} \right.$
 5. The computer method of claim 3, wherein the code comprises a (k+2, k) MDS array code, and the array has size 2^(m)×(k+2), wherein the permutations are defined by f_(j), jε[0,k−1] and for systematic information elements a_(i,j), and for row and zigzag parity elements r_(i) and z_(i), respectively, for iε[0,2^(m)−1], jε[0,k−1] and for row coefficients given by a_(i,j)=1 for all i, j, and zigzag coefficients given by β_(i,j) in some finite field F, the method further comprising: for a single erasure, (1) where one parity node is erased, rebuilding the row parity according to ${r_{i} = {\sum\limits_{j = 0}^{k - 1}\; a_{i,j}}},$ and rebuilding the zigzag parity by ${z_{i} = {\sum\limits_{j = 0}^{k - 1}\; {\beta_{{f_{j}^{- 1}{(i)}},j}{a_{{f_{j}^{- 1}{(i)}},j}.}}}};$ (2) where one information node j is erased, rebuilding the elements in rows X_(j) and those in rows X_(j) by zigzags; and for two erasures, (1) where two parity nodes are erased, rebuilding by the one parity erasure rebuilding, (2) where one parity node and one information node is erased, and if the row parity node is erased, then rebuild by zigzags; otherwise rebuilding by rows; (3) where two information nodes j₁ and j₂ are erased, then if f_(j) ₁ =f_(j) ₂ , for any iε[0,2^(m)−1], computing $x_{i} = {r_{i} - {\sum\limits_{{j \neq j_{1}},j_{2}}\; a_{i,j}}}$ ${y_{i} = {z_{f_{j\; 1}{(i)}} - {\sum\limits_{{j \neq j_{1}},j_{2}}\; {\beta_{{f_{j}^{- 1}{f_{j_{1}}{(i)}}},j}a_{{f_{j}^{- 1}{f_{j_{1}}{(i)}}},j}}}}};$ solving a_(i,j) ₁ , a_(i,j) ₂ from the equations ${\begin{bmatrix} 1 & 1 \\ \beta_{i,j_{1}} & \beta_{i,j_{2}} \end{bmatrix}\begin{bmatrix} a_{i,j_{1}} \\ a_{i,j_{2}} \end{bmatrix}} = \begin{bmatrix} x_{i} \\ y_{i} \end{bmatrix}$ else, for any iε[0,2^(m)−1], setting i′=i+f_(j) ₁ (0)+f_(j) ₂ (0), and computing x_(i), x_(i′), y_(i), y_(i′) according to the two information nodes operation (3) and then solving a_(i,j) ₁ , a_(i,j) ₂ , a_(i′,j) ₁ , a_(i′,j) ₂ from equations ${\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \beta_{i,j_{1}} & 0 & 0 & \beta_{i^{\prime},j_{2}} \\ 0 & \beta_{i,j_{2}} & \beta_{i^{\prime},j_{1}} & 0 \end{bmatrix}\begin{bmatrix} a_{i,j_{1}} \\ a_{i,j_{2}} \\ a_{i^{\prime},j_{1}} \\ a_{i^{\prime},j_{2}} \end{bmatrix}} = {\begin{bmatrix} x_{i} \\ x_{i^{\prime}} \\ y_{i} \\ y_{i^{\prime}} \end{bmatrix}.}$
 6. A controller of an array of storage nodes, the controller comprising: a host interface through which the controller communicates with a host computer; a node interface through which the controller communicates with the array of storage nodes; a processor that operates a controller application for: receiving configuration data at the controller that indicates operating features of the array; and determining a parity code for operation of the array according to a permutation, wherein the configuration data specifies the array as comprising nodes defined by A=(a_(i,j)) with size r^(m)×k for some integers k, m, and wherein for T={v₀, . . . , v_(k−1)}

Z_(r) ^(m) a subset of vectors of size k, where for each v=(v₁, . . . , v_(m))εT, gcd(v₁, . . . , v_(m), r), where gcd is the greatest common divisor, such that for any l, 0≦l≦r−1, and vεT, the code values are determined by the permutation f_(v) ^(l):[0, r^(m)−1]→[0, r^(m)−1] by f_(v) ^(l)(x)=x+lv.
 7. The controller of claim 6, wherein A specified by the configuration data comprises A=(a_(i,j)), an array of size 2^(m)×k for some integers k, m, and k≦2^(m), and wherein for T

F₂ ^(m) is a subset of vectors of size k that does not contain the zero vector, and for vεT the permutation is given by f_(v):[0,2^(m)−1]→[0,2^(m)−1] by f_(v)(x)=x+v, where x is represented in its binary representation.
 8. The controller of claim 7, wherein for the permutations f₀, . . . , f_(m) and for sets X₀, . . . , X_(m) constructed by the vectors {e_(i)}_(i=0) ^(m), the set X₀ is modified to be X₀={xεF₂ ^(m):x·(1, 1, . . . , 1)=0}.
 9. The controller of claim 7, further comprising generating an s-duplication code of the code values, such that generating the s-duplication code comprises assigning α_(i,j) ^((t))=1 for all i, j, t, such that for odd q, and s≦q−1 and assigning for all tε[0, s−1] $\beta_{i,j}^{(t)} = \left\{ \begin{matrix} {a^{t + 1},} & {{{if}\mspace{14mu} {u_{j} \cdot i}} = 1} \\ {a^{t},} & {o.w} \end{matrix} \right.$ where u_(j)=Σ_(l=0) ^(j)e_(l), and for even q (powers of 2), and s≦q−2 and assigning for all tε[0, s−1] $\beta_{i,j}^{(t)} = \left\{ \begin{matrix} {a^{{- t} - 1},} & {{{if}\mspace{14mu} {u_{j} \cdot i}} = 1} \\ {a^{t + 1},} & {{o.w}..} \end{matrix} \right.$
 10. The controller of claim 8, wherein the code comprises a (k+2, k) MDS array code, and the array has size 2^(m)×(k+2), wherein the permutations are defined by f_(j), jε[0,k−1] and for systematic information elements a_(i,j), and for row and zigzag parity elements r_(i) and z_(i), respectively, for iε[0,2^(m)−1], jε[0,k−1] and for row coefficients given by α_(i,j)=1 for all i, j, and zigzag coefficients given by β_(i,j) in some finite field F, the method further comprising: for a single erasure, (1) where one parity node is erased, rebuilding the row parity according to ${r_{i} = {\sum\limits_{j = 0}^{k - 1}\; a_{i,j}}},$ and rebuilding the zigzag parity by ${z_{i} = {\sum\limits_{j = 0}^{k - 1}\; {\beta_{{f_{j}^{- 1}{(i)}},j}{a_{{f_{j}^{- 1}{(i)}},j}.}}}};$ (2) where one information node j is erased, rebuilding the elements in rows X_(j) and those in rows X_(j) by zigzags; and for two erasures, (1) where two parity nodes are erased, rebuilding by the one parity erasure rebuilding, (2) where one parity node and one information node is erased, and if the row parity node is erased, then rebuild by zigzags; otherwise rebuilding by rows; (3) where two information nodes j₁ and j₂ are erased, then if f_(j) ₁ =f_(j) ₂ , for any iε[0,2^(m)−1], computing $x_{i} = {r_{i} - {\sum\limits_{{j \neq j_{1}},j_{2}}\; a_{i,j}}}$ ${y_{i} = {z_{f_{j\; 1}{(i)}} - {\sum\limits_{{j \neq j_{1}},j_{2}}\; {\beta_{{f_{j}^{- 1}{f_{j_{1}}{(i)}}},j}a_{{f_{j}^{- 1}{f_{j_{1}}{(i)}}},j}}}}};$ solving a_(i,j) ₁ , a_(i,j) ₂ from the equations ${\begin{bmatrix} 1 & 1 \\ \beta_{i,j_{1}} & \beta_{i,j_{2}} \end{bmatrix}\begin{bmatrix} a_{i,j_{1}} \\ a_{i,j_{2}} \end{bmatrix}} = \begin{bmatrix} x_{i} \\ y_{i} \end{bmatrix}$ else, for any iε[0,2^(m)−1], setting i′=i+f_(j) ₁ (0)+f_(j) ₂ (0), and computing x_(i), x_(i′), y_(i), y_(i′) according to the two information nodes operation (3) and then solving a_(i,j) ₁ , a_(i,j) ₂ , a_(i′,j) ₁ , a_(i′,j) ₂ from equations ${\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \beta_{i,j_{1}} & 0 & 0 & \beta_{i^{\prime},j_{2}} \\ 0 & \beta_{i,j_{2}} & \beta_{i^{\prime},j_{1}} & 0 \end{bmatrix}\begin{bmatrix} a_{i,j_{1}} \\ a_{i,j_{2}} \\ a_{i^{\prime},j_{1}} \\ a_{i^{\prime},j_{2}} \end{bmatrix}} = {\begin{bmatrix} x_{i} \\ x_{i^{\prime}} \\ y_{i} \\ y_{i^{\prime}} \end{bmatrix}.}$ 